4. Continuous Random Variables

Question 1

Expectation of a CRV

Answer 1

For a continuous random variable X, the expectation of a
function g(X) is defined as
E{g(X)} := (integral from ∞ to −∞) g(x)fX(x) dx.

Question 2

Variance of a CRV

Answer 2

Var(X) = E{(X - µ)^2}
= (integral from ∞ to −∞) (x - µ)^2 fX(x) dx.
Var(X) = E(X^2) - E(X)^2 still holds

Question 3

Exponential Distribution

Answer 3

Used to represent a ‘time to an event’.
pdf: fx(x) = λe^(-λx) when x >= 0.
E(X) = 1/λ
Var(x) = 1/(λ^2)

Question 4

Exponential distribution - memory loss property

Answer 4

If X~Exp(λ), then

P(X > x + a | X > a) = P(X > x)

Question 5

Uniform Distribution (CRV)

Answer 5

pdf: fx(x) = 1 / (b-a)
cdf: Fx(x) = (x-a) / (b-a)
E(X) = 0.5 * a * b
Var(x) = (b-a)^2 / 12

Question 6

Standard Normal Distribution

Answer 6

fz(z) = 1 / sqrt(2*pi) e^(-0.5z^2)

Question 7

Normal Distribution

Answer 7

fx(x) = 1 / sqrt(2piσ^2) e^{-1/(2σ^2) (x - µ)^2}

when the mean = µ and variance = σ

Question 8

If Z ~ N(0, 1) and X ~ µ + σZ, what is the distribution of X?

Answer 8

X ~ N(µ, σ)