IA: 1P4: Mathematics Flashcards
What is the triangle inequality for vector addition?
For vectors a and b:
What are the 2 forms for the equation of a vector line?
What is the standard form for the equation of a plane?
Where u and v are lines in the plane and a is a point in the plane
What is the scalar product between 2 unit vectors that are orthogonal to each other?
zero, 0
What is the scalar product between 2 unit vectors that are parallel to each other?
one, 1
What does the scalar product do?
The scalar product takes two vectors and returns a scalar result. It measures how much one vector “projects” onto another.
Is the scalar product distributive?
Yes
What is the equation for the scalar product using the angle between them?
Where θ is the outgoing angle between the 2 vectors
What is the equation for a plane in the form:
n = unit vector orthogonal to the plane
r = any point on the plane
d = shortest distance between the origin and the plane
What is the equation for the minimum distance between a point c and the line
Note: unit vector
Why does the order of the vector product matter?
It is non-commutative:
What is the result of the cross product between a vector and itself?
zero, 0
What is the equation for the vector product of 2 vectors using the angle between them?
θ = angle between a and b
n = unit vector normal to the plane containing both a and b
What is the magnitude of the cross product represented by?
The area of the parallelogram formed by the 2 vectors
How can the cross product be used to write a position vector?
If a x b ≠ 0, then any positon vector can be represented by:
What is the equation for the minimum distance between 2 skew lines, r₁ and r₂?
What is the equation for the scalar triple product?
What does it mean that the scalar triple product holds true for “cyclic permutations”?
- If you rotate the vector set “cyclically” the scalar triple product will remain true. “cyclic” means that the elements are rearranged in a cycle while keeping their order intact, i.e (a,b,c) & (c,a,b) & (b,c,a)
What are the 2 equations for the vector triple product?
What is the equation for the line of intersection of 2 planes?
c = point on the line of intersection, obtained by solving the two plane equations simultaneously
n = normal vectors of the 2 planes
What are the 4 scenarios of three planes intersecting?
- They don’t - three parallel planes
- 2 parallel planes with one plane intersecting both of them
- 3 planes intersecting in a prism or at a common line
- 3 planes intesect at a point
How can you determine if three planes are parallel?
The normal for each plane will be aligned
How can you determine if there are 2 parallel planes with one intersecting both of them?
The normal for 2 of the planes will be aligned, but not for all three
How can you determine if three planes form a prism / a common line of intersection?
When none of the planes are parallel, but there is no common point of intersection. This is the case if:
How can you determine if three planes all intersect at a point?
When none of the planes are parallel and solving the simultaneous equations gives a unique solution
How can you determine the coordinates of the point of intersection of all three planes?
How can these simultaneous equations be rewritten in matrix form?
What is an odd function?
What is an even function?
If a function f(x) is continuous at x = 0, what is true if f(x) is odd?
f(0) = 0
If a function f(x) is continuous at x = 0, what is true if f(x) is even?
f‘(0) = 0
What features do you need to consider when sketching a curve?
- Limits: What happens at x→±∞
- Intercepts: When does f(x) = 0, and likewise what is the value of f(0)?
- Symmetry: is f(x) even or odd (or neither), or is there a point about which it is symmetric
- Turning points: When f‘(x) = 0 what sign is f’‘(x) (maxima, minima, inflection)
- Singularities: Is there a value such that |f(x)|→±∞
What is sinh(x)?
What is cosh(x)?
What is tanh(x)?
What is the graph of sinh(x)?
Note: odd function
What is the graph of cosh(x)?
Note: even function
What is the graph of tanh(x)?
Note: odd function
What is cosh(x) + sinh(x)?
eˣ
What is cosh(x) - sinh(x)?
e⁻ˣ
what is cosh²(x) + sinh²(x)?
cosh(2x)
What is artanh(y) in logarithm form?
What is a taylor series?
A taylor series for a function f(x) is a polynomial expansion about a point x = a such that:
What is the formula for the taylor series expanded about x = a?
Where f⁽ⁿ⁾(a) is the n’th derivative of f(x) evaluated at a
What is the maclaurin series?
It is the taylor series expanded about x = 0
What is the radius of convergence of a taylor series?
The radius of convergence of a Taylor series is the distance from the center of the series (the expansion point) to the nearest point where the series fails to converge
How do you approach a taylor expansion for a large x, x→∞
A common approach is to let y = 1/x and then consider a series for which y is small, y→0.
What is L’Hôpital’s rule?
When can L’Hôpital’s rule be used?
For indeterminate cases where the limit produces 0/0 or ∞/∞
How can you evaluate a limit when L’Hôpital’s rule cannot be used?
You can use a series expansion (such as taylor or maclaurin) to approximate a function in polynomial form and use this to approximate the limit
What is the sum of a complex number (z) and its conjugate?
2 Re(z)
What is the difference between a complex number (z) and its conjugate?
2j Im(z)
What loci in the complex plane does this represent?
A circle of radius k centred at the complex number z₁
What loci in the complex plane does this represent?
An ellipse
For unfamiliar forms, how can you determine the loci formed in the complex plane?
Let z = x + jy and then expand to form a cartesian equation
What is Eulers formula?
What is the exponential form of cos(x)?
What is the exponential form of sin(x)?
How can Euler’s formula be used for evaluating integrals with trigoneometry?
Example:
Rather than using integration by parts, you can replace the sin with an exponential using Euler’s formula and just take the imaginary component at the end of the workings
What is cos(jy)?
cosh(y)
What is sin(jy)?
j sinh(y)
What is cosh(jy)?
cos(y)
What is sinh(jy)?
j sin(y)
How can you determine the value of a function such as arcsin(2) (when sin is normally limited to between 1 and -1)?
Let:
x + jy = arcsin(2)
Solve using compound angle formula and hyperbolic functions
Derive the complex impedance of a resistor:
Derive the complex impedance of a capacitor:
Derive the complex impedance of an inductor:
What is an ordinary differential equation?
A differential equation where the dependent variable is only a function of the single independent variable
What is a partial differential equation?
A differential equation which has more than one independent variable
What are the 4 methods for solving a first order linear differential equation?
- Distinct Integration
- Separable Equations
- Integrating Factor method
- Equations reducable to the separable form (scaling)
Solve this differential equation using the “scaling” method
y = x arctan(Ln|x| + c)
What is a linear ODE?
An ordinary differential equation in which the dependent variable (such as y) and its derivatives only appear as a linear combination.
What is a homogenous ODE?
An ordinary differential equation which has no functions of the independent variable appearing on its own (or simply the right hand side of the equation is zero)
What are the 3 cases for the solution of an auxiliary equation for a second order ODE?
- Real and distinct roots
- Complex conjugate roots
- Repeated real roots
If the auxiliary equation for a second order homogeneous ODE produces 2 real and distinct roots, λ₁ and λ₂, what is the general solution of the differential equation?
If the auxiliary equation for a second order homogeneous ODE produces complex conjugate roots, α±iβ, what is the general solution of the differential equation?
If the auxiliary equation for a second order homogeneous ODE produces a single repeated root, α, what is the general solution of the differential equation?
How do you solve a non-homogenous second order ODE?
- Find the general solution of the homogenous equation (known as the complementary function)
- Determine the particular integral
- The general solution is the complementary function + particular integral
How can you determine the trial particular integral to begin with?
Trial and error, however there is a table of common particular integrals in the data booklet:
What are “troublesome” cases with particular integrals?
When the right hand side of the differential equation has the same form as part of the complementary function. In these cases an alternative PI should be used. These also appear in a table in the data booklet:
What is modelling with differential elements?
- Draw a diagram of an element
- Balance forces/moments or energy fluxes to get an ODE
- Take δx → 0 to obtain an ODE
- Solve the ODE
What is a linear difference equation?
A type of recurrence relation that relates the values of a sequence at different points, with the relationship being linear
How do you find the general solution of a linear homogeneous difference equation?
Substitute yₙ = Aλⁿ:
This will form a quadratic equation that functions in the same way as an auxiliary equation. Therefore you can then continue by solving in a similar way to a second order linear ODE:
We are looking for a solution in the form yₙ = Aλⁿ
How do you solve a non-homogeneous linear difference equation?
- Find the “complementary function” by solving with the RHS as 0
- Find a “particular solution” this is analagous to a particular integral
- General solution = complementary function + particular solution
- Determine constants from intial conditions
What is the partial derivative of f(x,y) with respect to x defined as?
What is the partial derivative of f(x,y) with respect to y defined as?
For a function f(x,y), what does “partial differentiation with respect to x” mean?
For partial differentiation with respect to x, y is treated as a constant and you are slicing through the surface parallel with the x axis. Therefore, when differentiating, you do it completely as normal whilst treating y as a constant.
For a function with more than one independent variable f(x,y), how can you estimate the change in the output of the function (δf) if there is a small change in both the independent variables (δx and δy)?
What is a total differential?
The value of the small change of a function when the small changes in the independent variables tend to zero
What is linearisation (in reference to partial derivatives)?
A method of approximating a multivariable function near a given point using the first-order terms of its Taylor series expansion
What is the equation for the linearisation estimate about the point (a,b) for the function f(x,y)?
What is the directional derivative for a multivariable function?
The gradient along a particular direction: It is the rate of change of the function f along the direction u = (a,b), where u is a unit vector
What is the equation for the directional derivative?
∇f = ∇f(x,y)
u = (a,b)
What does “∇f(x,y)” mean?
The gradient of a multivariable function f(x,y)
What is the chain rule for partial derivatives?
How can a transformation be written in terms of matrices?
p’ = transformed coordinates
A = matrix
p = original coordinates
What is an identity matrix?
An identity matrix, I, is defined such that for a square matrix A:
Is matrix multiplication associative?
Yes:
Is matrix multiplication commutative?
No, the order is important
Is matrix multiplication distributive over addition and subtraction?
Yes:
What is the transpose of a matrix?
The transpose of a matrix is formed by intercahnging the rows and columns of a matrix
For matrices A and B, what is the transpose of (AB)?
How can the dot product of 2 vectors be expressed using matrix multiplication?
What is an inverse matrix?
For a square matrix A, the inverse A⁻¹ is defined such that:
A⁻¹A = AA⁻¹ = I
How do you calculate the inverse of a 3x3 matrix?
- Find the determinant of A, Det A
- Form the matrix of minors, M
- From the matrix of minors, form the matrix of cofactors, C
- Find the Transpose of the matrix of cofactors, Cᵀ
- A⁻¹ = (1/DetA) Cᵀ
What is the adjugate matrix?
The transpose of the cofactor matrix
When does the determinant of a square matrix change sign?
When 2 rows (or columns) are exchanged
When is the determinant of a square matrix zero (0)?
If two rows (or columns) are equal
How can you simplify a matrix in order to make it easier to find the determinant?
You can add a multiple of one row to another row (or one column to another) as it will leave the determinant unchanged
How does the determinant of a matrix and its transpose differ?
They do not, they are the same
What are the 3 main transformations performed by matrices?
- Rotations
- Reflections
- Pure stretches/compressions
How can you determine the matrix for a transformation?
Consider the columns of the transformation vector (a1, a2, a3). These are the vectors that result from transforming the x- y- and z- unit basis vectors respectively. So to determine the matrix, all you must establish is where the three basis vectors map to under the transformation.
What direction does a rotation matrix act?
anticlockwise
What is the rotation matrix in two-dimensions?
What is an orthogonal matrix?
A matrix with the property:
What is an orthogonal matrix with a determinant of one known as?
- A proper orthogonal matrix
- A rotation matrix
What is the matrix for a rotation about the y-axis in 3-dimensional space?
What is an orthonormal set of vectors?
Unit vectors that are mutually perpendicular
Are rotation matrices commutable?
No, rotations do not commute in 3D unless they are about the same axis. Therefore rotations are commutable in 2 dimensions
What is the determinant of a reflection matrix?
-1
If the coordinate frames are rotated by a matrix R but the point a is not rotated, what is the expression for the position of point a in the new coordinate frames?
It has the same effect as keeping the coordinate frame fixed, but rotating the vector in the opposite direction. Therefore:
Rᵀ = R⁻¹ as it is a pure rotation
If a matrix, A, transforms a position vector b to position c but then the coordinate frames are rotated by a matrix R, what is the expression for the new matrix which will transform position vector b to position vector c?
A’ = Rᵀ A R
How can rotating a coordinate frame help you to determine the matrix for the shear along an axis?
You can find the transformation in a convenient coordinate system that makes the shear transformation simple. You can then transform it back to the orignal coordinates. All you require is the orthognal matrix R that describes the rotation of the original coordinate system to the new coordinate system: e’ = Re.
A’ = RᵀAR
What is an eigenvector?
An eigenvector is a special vector associated with a square matrix (or a linear transformation) that remains aligned in the same direction after the transformation, although it may be scaled (stretched or compressed). Where A is the transformation, x is a non-zero vector, and λ is a scalar: x is said to be an eigenvector if it satisfies the equation:
What is λ in this equation for an eigenvector?
It is a scalar value known as an eigenvalue
What is an eigenvalue, λ?
The scale factor applied to the magnitude of the eigenvector following the transformation
Can an eigenvalue be zero?
Yes, the only important thing is that the eigenvector is non-zero
Does the magnitude of an eigenvector matter?
No, only the direction. If x is an eigenvector of A, then any scalar multiple of x is also an eigenvector of A. It is customary (but not obligatory) to normalise the eigenvectors so that they are unit vectors anyway.
How can you find the eigenvectors and eigenvalues of a matrix A?
A final (optional) step can then be to normalise the eigenvectors to make them unit length
How many eigenvalues with a matrix have?
For a nxn matrix, there will be exactly n eigenvalues.
When is a matrix symmetrical?
A = Aᵀ
What is true about the eigenvectors and eigenvalues of any symmetric matrix?
- The eigenvalues are real
- The eigenvectors are orthogonal
What is an antisymmetric matrix?
Aᵀ = -A
What is a defective matrix?
A defective matrix is an nxn matrix which has fewer than n linearly independent eigenvectors. This may occur if the matrix has a repeated eigenvalue
If the 3x3 matrix A has the eigenvalues and eigenvectors given below, how can they be combined into a single matrix equation?
Where u₁, u₂, and u₃ are of UNIT LENGTH!!!
A = U Λ U⁻¹
What is the condition for the equation “A = U Λ U⁻¹” to be formed from a matrix and its eigenvectors and eigenvalues?
Where u₁, u₂, and u₃ are of UNIT LENGTH!!!
For U⁻¹ to exist, det(U) ≠ 0. For a 3x3 matrix, this requires u₁, u₂, and u₃ to be linearly independent. Therefore defective matrices cannot be diagonalised as they do not have a full set of linear independent eigenvectors.
What can be said when diagonalising a symmetric matrix?
A symmetric matrix has orthogonal eigenvectors so they are linearly independent and so U⁻¹ always exists. Furthermore, U⁻¹ = Uᵀ and so for a symmetric matrix S:
For a real symmetric matrix, what does the diagonal matrix Λ represent?
It represents the same physical transformation as S but in a basis (coordinate system) aligned with the orthogonal eigenvectors. This is because:
What physical transformation do all symmetric matrices represent?
For any symmetric matrix, the transformation it represents is a pure stretch (or compression) along the mutually orthogonal eigenvectors with the scale factor being the eigenvalues (which are always real). There is no rotation or skewing as they are scaling along orthogonal axes.
What is the product of all the eigenvalues of a matrix equal to?
The determinant of the matrix, and subsequently the scale factor of the transformation
For the matrix A with linearly independent eigenvectors (but not necessarily orthogonal), what is Aⁿ given by?
For the matrix A with linearly independent eigenvectors, what is Aⁿx given by as n→∞
u₁ u₂ u₃ are normalised eigenvectors
Note: if λ₂ was the largest eigenvalue, it would be α₂ and u₂ etc etc
What is an eigenplane?
An eigenplane of a matrix or linear transformation is a two-dimensional subspace (a plane) on which any vector gets mapped to another point on this plane following the linear transformation. An example of where an eigenplane may occur is a 3D rotation
When does an eigenplane occur (for a 3x3 matrix)?
- When there is a complex conjugate pair of eigenvalues
- When two of the eigenvectors are complex
What is a linear system?
In a linear system the output is computed as some linear combination of the inputs
What are the properties of a linear system?
- Linear systems satisfy the principle of superposition (see image)
- A linear time invariant system have the property that a sine wave at the input leads to a sine wave at the output, with the same frequency (However the amplitude and/or phase can change)
What is a time invariant system?
A system is time invariant if delaying an input results in the same output, just delayed (by the same amount): if f(t) → y(t), then f(t - τ) → y(t - τ) for any τ
What is a step function?
A step function is a piecewise constant function that changes values only at specific points, creating a series of flat segments or “steps”
What is the Heaviside step function [H(t)]?
The unit step function where the step is at t = 0
What is the Dirac Delta function (impulse function)?
The delta function is a spike with unit area, it is infinitely tall but infinitely thin. δ(x-a) is zero everywhere apart from at a. The definite integral of δ(x-a), where the bounds pass over x = a will equal 1, if the bounds do not pass over x = a it will equal zero.
How are the delta function and step function related?
- The integral of the delta function is the step function
- The derivative of the step function is the delta function
What is the sifting theorem?
When you integrate a function multiplied by a delta function, the result is simply the function’s value at the point where the delta function is centered.
If “b” is not within the integration range, then the output is zero.
How can you determine the output for a differential equation for any given input?
Using convolution
For the step where you solve to find the step response, how do you do this?
Set the input f(t) = H(t). However since this cannot be done directly we must:
What does r(t) usually represent?
step response
What does g(t) usually represent?
impulse response
If an input to a linear system is composed of many impulses, how can you find the corresponding output?
Solve the differential equation to find the impulse response (by differentiating the step response) and then use superposition to find the corresponding output:
How does convolution work?
Considering the input f(t) to be made up of a sequence of strips of width Δτ, each strip can be formed from a scaled and delayed delta function. Therefore the output is the sum of these delated, scaled impulse responses. As Δτ approaches zero, this sum turns into an integral called the convolution integral.
What is the convolution integral?
- Treat t as a constant, τ is the integration variable
- t is the time as it relates to the output of the system y(t)
- τ is the time as it relates to the input of the system f(t)
What is the purpose of convolution?
It allows you to find the output of a differential equation from the input f(t) once you have determined the impulse response
When evaluating convolution integrals, what must you remember about the inputs?
You may have to split up the integrals if there is a piecewise function for an input (i.e a series of step functions). However, when splitting them up you must remember that the original convolution integral starts from -∞ and so you must include all previous stages aswell. By including the previous stages over their entire ranges (rather than up to time t) it gives a constant as the start point for the stage! For example:
Note: For the second and third stages the stages previous have still been included over their entire range (rather than up to t) as to give a constant!!!
If the expression for g(t) is very complicated, what can you do to make the convolution integral easier?
If the expression for g(t) is very complicated, it may be easier to compute the integral if it has a factor in the form g(τ) rather than g(t-τ). Therefore you can swap the arguments to the functions in the convolution integral to make it easier to compute. It does not matter which way round the arguments to the functions in the convolution integral are, so long as both functions are zero for t < 0
Only valid when both functions are zero for t < 0!!! (This is almost always the case for time-varying systems)
What is a time varying system?
These are systems which have a temporal input (time), they have no output before the input that causes it and so g(t) = 0 for t < 0
What is a spatially varying system?
These are systems which respond to spatial inputs, the inputs can affect the output on either side and therefore g(x) can be non-zero for any x (including x < 0)
How do systems responding to a temporal input work?
Only “past” inputs contribute to the output at t
How do systems responding to a spatial input work?
Inputs to both the left and right of x contribute to the output at x
What is the spatial convolution integral?
Note: upper bound is not x
What are causal systems?
Systems for which g(t) = 0 for all t < 0
What type of systems are always causal?
Systems with time-varying inputs and outputs
This example has no governing differential equation, explain how convolution is still used to solve the displacement under a constinuous load K
IMPORTANT EXAMPLE
If you can determine the spatial impulse response, you can compute the complete displacement under a continuous load K using the spatial convolution integral. To find this impulse response you first have to determine the maximum displacement for a point load, F = 1, at position a. This lets you express the impulse response as two piecewise functions, corresponding to the two straight-line segments of the response. You can then use the convolution integral (split into a sum for the 2 sections) to find the overall displacement as a function of x.
What are the basis functions for a fourier series?
What is a fourier series?
A Fourier series represents a periodic function as an infinite sum of sines and cosines with different frequencies and amplitudes.
What is the general expression of a fourier series over a length of 2π?
For a length of 2π:
What is the expression for aₙ for a fourier series with a length of 2π?
Note: the bounds do not have to be +-π, they can be any combindation with a length of 2π. f(t) is only modelled within this range and so will repeat with a period of 2π outside this range and thus will only be useful over this range (unless f(t) is periodic itself)
What is the expression for bₙ for a fourier series with a length of 2π?
Note: the bounds do not have to be +-π, they can be any combindation with a length of 2π. f(t) is only modelled within this range and so will repeat with a period of 2π outside this range and thus will only be useful over this range (unless f(t) is periodic itself)
What is the expression for d for a fourier series with a length of 2π?
Note: the bounds do not have to be +-π, they can be any combindation with a length of 2π. f(t) is only modelled within this range and so will repeat with a period of 2π outside this range and thus will only be useful over this range (unless f(t) is periodic itself).
What is true about the fourier series of a function if it is an even function?
bₙ = 0
The aₙ terms model the even component and the bₙ terms model the off component in the function. The d term models the mean value of the function.
What is true about the fourier series of a function if it is an odd function?
aₙ = 0
The aₙ terms model the even component and the bₙ terms model the off component in the function. The d term models the mean value of the function.
What is true about the fourier series of a function if it has a mean value of zero?
d = 0
The aₙ terms model the even component and the bₙ terms model the off component in the function. The d term models the mean value of the function.
What is the general expression of a fourier series over a general range?
What is the expression for aₙ for a fourier series over a general range?
What is the expression for bₙ for a fourier series over a general range?
What is the expression for d for a fourier series over a general range?
What is the fundamental angular frequency?
the fraction 2π/L, often written as ω₀. It appears in the expression for a fourier series:
What are the 3 ways to find the Fourier series for f(x) between 0 and L?
- Use the general range Fourier formulae directly
- Differentiate the waveform twice to get a sequence of delta functions. Find a Fourier series for the delta functions, and then integrate the series twice to get the Fourier series of the triangular wave.
- Look up the Fourier series of a similar waveform in the Maths Data book and use a substitution of variables to find the series for the waveform we require
How is a triangle wave related to step functions and delta functions?
Why must you be careful when finding the Fourier series of a function containing delta functions?
You must choose your bounds carefully as to ensure the limit does not lie on a delta function as it is uncertain what to do in that scenario. In the below example rather than choosing bounds of 0 and L, you may choose -L/4 and 3L/4. The fourier series will still have a period of L, but the integral becomes much clearer.
What is the convergence of a Fourier series?
The convergence of a Fourier series describes how closely the series approximates the original function as more terms are added, i.e. a square wave converges with 1/n and a triangular wave converges with 1/n²
What is the fourier convergence of a function which is a series of delta functions?
It does NOT converge
What is the fourier convergence of a function which has a discontinuous value, such as a square wave?
Converges as 1/n
What is the fourier convergence of a function which has a discontinuous gradient, such as a triangular wave?
Converges as 1/n²
What is the fourier convergence of a function which has a discontinuous second derivative?
Converges as 1/n³
What is a “Half range” series?
If you want to model a signal f(x) in the range 0 to T, you can use the fourier formulae for a general series to generate a variery of different serieses. They will all be the same in the range 0 to T, but since this is the only part we care about they may differ outside this range. Different serieses converge at different rates and therefore may be suitable for different scenarios.
Why may you use a “Half range” series?
- Some series may converge faster than others
- Some series are easier to calculate (i.e. some of aₙ, bₙ, or d are zero)
- You may be limited by the demands of the questions (i.e. it must be even or odd or have a given convergence)
What are the basis functions for a complex fourier series?
These can all be represented by: eʲⁿᵗ
j = i
What is the general complex fourier series for a series of length 2π
Note: Lower bound is -∞, not 1
What is the general complex fourier series for a series of general length, L
Note: Lower bound is -∞, not 1
What is the expression for Cₙ for a complex fourier series of length 2π
Note: bounds do not have to be between 0 and 2π, they can be any combination as long as the length is 2π
What is the expression for Cₙ for a complex fourier series of general length L
When must you be careful when producing a complex fourier series?
Cₙ may have an undefined value for a given n, therefore you must take the limit for this value. Notice in the below example the fourier series has been split into 3 sections: -∞ to -1, 0, and 1 to ∞. This ensures that the undefined value is accounted for
Remember to look out for this!
How can you convert a complex fourier series (in terms of e) into a real fourier series (in terms of sines and cosines)?
You can determine the real coefficients aₙ, bₙ, and d from the complex coefficient cₙ:
* aₙ = 2 Re(Cₙ)
* bₙ = -2 Im(Cₙ)
* d = c₀
These are then the coefficients you can use for the normal Fourier series
How can you convert a real fourier series (in terms of sines and cosines) into a complex fourier series (in terms of e)?
You can determine the complex coefficient Cₙ from the real coefficients aₙ and bₙ:
What is the expression for P(A or B) when A and B are mutually exclusive?
What is the expression for P(A or B) when A and B are not mutually exclusive?
What is the expression for P(A and B) when A and B are independent?
What is the expression for P(A and B) when A and B are not independent?
What is the number of different orders in which n unique objects can be placed?
n! (n factorial)
What is the number of ways of choosing r items from n when the order of the chosen items matters?
What is the number of ways of choosing r items from n when the order of the chosen items does not matter?
What is the equation for the arithmetic mean of a population?
What is the equation for the variance of a population?
What is the equation for the standard deviation of a population?
What is the equation for the estimate of the arithmetic mean based on a sample of a population?
What is the equation for the estimate of the standard deviation based on a sample of a population?
Give both forms of the equation
What is a discrete probability distribution?
A probability distribution where each event can only carry certain integer values for the probability
What is a uniform probability distribution?
Where each event has the same probability
What is the equation for the arithmetic mean of a population, based on the probability distribution?
What is the equation for the variance of a population, based on the probability distribution?
What is the equation for the standard deviation of a population, based on the probability distribution?
For a continuous probability distribution with a probability density function fx(x), what is the probability of a ≤ x ≤ b?
For a continuous probability distribution with a probability density function fx(x), what is the expression for the mean?
For a continuous probability distribution with a probability density function fx(x), what is the expression for the standard distribution?
What is a sample mean?
A sample mean is the mean average of a sample, it is also a random variable as many samples can be taken from the population
What is the mean of a set of sample means?
μ
What is the standard deviation of a set of sample means?
What is a gaussian distribution?
The normal distribution
What is the equation for the gaussian/normal distribution?
μ = mean
σ = standard deviation
What is the central limit theorem?
Regardless of the original distribution of a population, the distribution of the sample means (or sums) approaches a normal (Gaussian) distribution as the sample size increases, provided the samples are independent and identically distributed
For a Gaussian/normal distribution, what is the “empirical rule”?
- 50% of the data falls within 0.67σ of μ
- 68% of the data falls within 1σ of μ
- 95% of the data falls within 2σ of μ
- 99.73% of the data falls within 3σ of μ
How many times must an experiment be performed to have sufficient data to use the central limit theorem?
~30 times
