Topic 2B Random Variables Flashcards
What is a random variable?
A quantity whose values result from a random experiment.
Definition Recall
A random variable is a mapping from the sample space Ω to a subset of:
A. Integers
B. Real numbers
C. Both A and B
C. Both A and B (Z for discrete, R for continuous)
Discrete variables map Ω to __, while continuous variables map Ω to __.
Z (integers), R (real numbers)
Give one example each of a discrete and continuous random variable.
Discrete: number of marbles picked.
Continuous: wave heights in metres.
CDF is defined as: F(x) = __
P(X ≤ x)
CDF is valid for both discrete and ______ variables.
continuous
Formula
p(x) = P(X = x)
Total: ∑p(x) = __
1
Formula Recall
f(x) =
dF(x)/dx (derivative of the CDF)
The total area under the PDF must equal __.
1
Discrete: uses PMF or PDF?
Continuous: uses PMF or PDF?
Discrete → PMF
Continuous → PDF
PMF: p(x) = P(X = x)
PDF: f(x)dx = __
P(x ≤ X < x + dx)
What does the steepest part of the CDF represent?
The peak (mode) of the corresponding PDF
Formula Recall
For continuous X:
P(a ≤ X ≤ b)
∫from a to b of f(x) dx
For continuous X, P(X = a) = __
0
What is the quantile function?
It is the inverse of the CDF, mapping probabilities back to values of X.
Formula
qₚ(X) = u such that F(u) =
p%
Definitions
Discrete moment: E(Xᵐ) =
Continuous moment: E(Xᵐ) =
∑ xᵐp(x)
∫ xᵐf(x) dx
Centred moment =
E((X − E(X))ᵐ)
Mean: µ =
Variance: Var(X) =
Standard deviation: σ =
Skewness: g₁(X) =
E(X)
E(X²) − (E(X))²
√Var(X)
E((X − µ)³) / σ³
In physical analogy, what does the variance represent?
Moment of inertia (spread of distribution)
The mean is analogous to the ______ of mass
centre
What is the median of a random variable X?
The value such that P(X ≤ median) = 0.5
The mode is the value where the ___ or ___ is maximum.
PMF or PDF
What kind of functions h(X) are considered in this topic?
Monotonic, univariate functions
If Y = h(X) and h is invertible:
Discrete: p_Y(y) = p_X(h⁻¹(y))
Continuous: f_Y(y) = f_X(x) × |d/dx h(x)|⁻¹
Formula
E(h(X)) =
∑ h(x)p(x) (discrete)
∫ h(x)f(x) dx (continuous)
E(a ± bX) = ____
Var(a ± bX) = ____
a ± bE(X)
b²Var(X)
When does Var(X ± Y) = Var(X) ± Var(Y) hold?
Only if X and Y are independent
How do you find F(x) from f(x)?
Use dummy variables
