FMAT prob&stat continuous random variables incompl Flashcards
What is a continuous random variable?
A continuous random variable is a variable that can take any real value within a given range. Unlike discrete random variables, which take specific values, a continuous random variable can assume infinitely many values within an interval. Examples include the weight of a newborn baby, the lifetime of a light bulb, or the time taken to swim a length of a swimming pool.
How is the probability of a continuous random variable determined?
The probability of a continuous random variable falling within a certain range is determined using a probability density function (p.d.f.), denoted as f(x). The probability of the variable taking a specific value is always zero, but the probability of it falling within an interval is found by integrating f(x) over that interval.
What is the key property of a probability density function (p.d.f.)?
The total area under the probability density function must be equal to 1. This ensures that the sum of all possible probabilities for the variable equals 1, representing certainty that some outcome will occur.
How do you calculate the probability that a continuous random variable falls within a given interval?
The probability that a continuous random variable X falls within the interval [a, b] is given by the integral:
P(a ≤ X ≤ b) = ∫[a to b] f(x) dx.
This integral computes the area under the curve of f(x) from x = a to x = b.
What is a piecewise-defined probability density function?
A piecewise-defined probability density function is a function that has different expressions for different ranges of x. This means that f(x) is defined by separate formulas depending on which part of the domain x falls into.
How do you determine unknown constants in a piecewise-defined p.d.f.?
To determine unknown constants in a piecewise-defined function, we use the condition that the total probability must be 1. This means integrating f(x) over all possible values of x and setting the result equal to 1:
∫ f(x) dx = 1.
If the function has different expressions in different intervals, the integral is computed separately for each interval, and their sum must equal 1.
What is the expectation (expected value) of a continuous random variable?
The expectation (or mean) of a continuous random variable X, denoted as E(X), is the weighted average of all possible values of X. It is given by:
E(X) = ∫ x * f(x) dx.
This integral finds the balance point of the distribution, representing the long-run average value of X over many repetitions.
What is the expectation of a function of a continuous random variable?
If Y is a function of X, written as Y = g(X), then the expectation of Y is found by integrating g(X) multiplied by the probability density function of X:
E(Y) = ∫ g(x) * f(x) dx.
This generalizes expectation beyond just X itself, allowing us to compute expected values for transformed variables.
What is the expectation of a linear function of a random variable?
If Y is a linear function of X, given by Y = aX + b, then its expectation follows the rule:
E(aX + b) = aE(X) + b.
This means scaling a random variable by a factor a scales its expectation by a, and adding a constant b shifts the expectation by b.
Example: If a random variable X has an expectation of 5, what is the expectation of Y = 2X + 3?
Using the linearity property of expectation:
E(Y) = 2E(X) + 3 = 2(5) + 3 = 10 + 3 = 13.
How do you calculate the probability that a function of a random variable falls below a given value?
To find P(g(X) ≤ c) for some function g(X), integrate the probability density function f(x) over the values of x that satisfy g(x) ≤ c.
What is the meaning of a probability density function defined in terms of a polynomial or cubic function?
When a probability density function is expressed as a polynomial or cubic function, it means the probability distribution follows a curved shape rather than a simple linear or uniform distribution. The area under this function still must sum to 1, and probability calculations require integrating polynomial expressions.
How can the probability of an event be found using an alternative approach?
Instead of directly computing P(A), we can use the complement rule:
P(A) = 1 - P(A’).
This is useful when P(A’) (the probability of the event not happening) is easier to compute than P(A).
Why does expectation behave linearly for linear functions of a random variable?
Expectation follows linearity because integration (which defines expectation) is a linear operation. This means that scaling a variable scales its expected value proportionally, and adding a constant shifts the expected value without affecting its variability.
What is the cumulative distribution function (CDF) of a continuous random variable?
The CDF, denoted as F(x), gives the probability that a continuous random variable X is less than or equal to a given value x. It is found by integrating the probability density function (PDF) from the lower limit up to x:
F(x) = ∫ f(t) dt (from lower limit to x).
Why is the CDF useful in probability calculations?
The CDF allows multiple probabilities to be calculated without repeating integrations. For example, to find P(a ≤ X ≤ b), simply compute F(b) - F(a) instead of integrating the PDF twice.
What is the key property of a CDF for a continuous random variable?
For a continuous random variable X with range [a, b]:
- F(a) = 0 (no probability mass below a).
- F(b) = 1 (all probability mass is at or below b).
- F(x) is a non-decreasing function.
How do you calculate the median using the CDF?
The median m of X is the value satisfying F(m) = 0.5, meaning that there is a 50% probability of X being less than or equal to m.
Example: A random variable X has PDF f(x) = (3/21)(x - 1) for 1 ≤ x ≤ 4. Find its CDF.
F(x) = ∫(3/21)(u - 1) du (from 1 to x).
Evaluating this integral gives:
F(x) = (3/21) * ((x - 1)² / 2) = (3/42) * (x - 1)², for 1 ≤ x ≤ 4.
How do you compute probabilities using a CDF?
To find P(a ≤ X ≤ b), use:
P(a ≤ X ≤ b) = F(b) - F(a).
Example: Using F(x) = (3/42) * (x - 1)², for 1 ≤ x ≤ 4 from the previous example, find P(X ≤ 2).
P(X ≤ 2) = F(2) = (3/42) * (2 - 1)² = 3/42 = 1/14.
Example: Using the same CDF F(x) = (3/42) * (x - 1)² for 1 ≤ x ≤ 4, find P(3 < X < 4).
P(3 < X < 4) = F(4) - F(3).
F(4) = (3/42) * (4 - 1)² = (3/42) * 9 = 27/42.
F(3) = (3/42) * (3 - 1)² = (3/42) * 4 = 12/42.
So, P(3 < X < 4) = 27/42 - 12/42 = 15/42 ≈ 0.357.
How do you find the probability density function (PDF) from a CDF?
The PDF is the derivative of the CDF:
f(x) = d/dx [F(x)].
Example: If X has CDF F(x) = x²/64 for 0 ≤ x ≤ 8, find the PDF
Differentiate F(x):
f(x) = d/dx (x²/64) = (2x/64) = x/32, for 0 ≤ x ≤ 8.
How do you find the PDF of a transformed variable Y = g(X)?
- Find the CDF of Y: G(y) = P(Y ≤ y) = P(g(X) ≤ y).
- Express G(y) in terms of F(x).
- Differentiate G(y) to get the PDF g(y).
Example: If Y = 3X and X has CDF F(x), how do you find the CDF of Y?
Use the transformation:
G(y) = P(Y ≤ y) = P(3X ≤ y) = P(X ≤ y/3) = F(y/3).
To find the PDF, differentiate G(y):
g(y) = (1/3) * f(y/3).
How do you find the PDF of U = 1 - X if X has a known CDF?
Use the transformation:
H(u) = P(U ≤ u) = P(1 - X ≤ u) = P(X ≥ 1 - u).
Since P(X ≥ a) = 1 - F(a), we get:
H(u) = 1 - F(1 - u).
The PDF of U is then found by differentiating H(u).
Why does the order of inequalities matter when transforming CDFs?
The function defining the transformation may reverse the order of inequalities. If X increases but the transformation Y = g(X) decreases, the CDF must be adjusted to account for this reversal.
What are common applications of CDF transformations?
- Finding the probability distribution of scaled and shifted variables (e.g., Y = 2X + 5).
- Transforming variables in statistics and physics.
- Deriving new probability distributions from known ones.
