2.16.1 Shifting and Scaling

Question 1

What does it mean to shift a random variable X by a constant c?

Answer 1

Define Y = X + c. Y is a shifted version of X. The distribution is moved left/right on the number line.

Question 2

What does it mean to scale a random variable X by a constant a?

Answer 2

Define Y = aX. Y is a scaled version of X. The distribution is stretched or compressed.

Question 3

What’s the effect of shifting X on the mean and variance?

Answer 3

E[X + c] = E[X] + c, Var(X + c) = Var(X)

Question 4

What’s the effect of scaling X on the mean and variance?

Answer 4

E[aX] = a · E[X], Var(aX) = a² · Var(X)

Question 5

What happens when a random variable is both shifted and scaled?

Answer 5

Y = aX + c. Apply scaling first, then shifting: do aX, then add c.

Question 6

If X ~ Normal(μ, σ²), what is the distribution of aX + c?

Answer 6

aX + c ~ Normal(aμ + c, a²σ²)

Question 7

If X ~ Uniform(a, b), what is the distribution of X + c?

Answer 7

X + c ~ Uniform(a + c, b + c)

Question 8

If X ~ Uniform(a, b), what is the distribution of aX?

Answer 8

aX ~ Uniform(a·a, a·b) (if a > 0); scale both endpoints

Question 9

If X ~ Exponential(θ), and Y = aX for a > 0, what is the distribution of Y?

Answer 9

Y ~ Exponential(aθ)

Question 10

If X ~ Exponential with rate λ, and Y = aX for a > 0, what is the distribution of Y?

Answer 10

Y ~ Exponential with rate λ’ = λ / a

Question 11

Does a shifted exponential random variable still follow an exponential distribution?

Answer 11

No. X + c (with c ≠ 0) is not exponential anymore. It’s called a shifted exponential.

Question 12

What’s the main difference between exponential and shifted exponential distributions?

Answer 12

Their range: Exponential starts at 0, shifted exponential starts at the shift amount.

Question 13

If X ~ Exponential(θ), and Y = X + c, what are the mean and variance of Y?

Answer 13

E[Y] = θ + c, Var(Y) = θ²

Question 14

How is the memoryless property used with shifted exponentials?

Answer 14

If X ~ Exp(θ), then X - t | X > t ~ Exp(θ). This implies X | X > t ~ t + Exp(θ)

Question 15

What’s an example of scaling an exponential variable?

Answer 15

If X ~ Exp(4), and Y = 2X, then Y ~ Exp(8)

Question 16

When shifting and scaling a normal variable, does it remain normal?

Answer 16

Yes. The normal distribution is closed under linear transformations.

Question 17

In example 2.16.2, why is aX + c still normal if X ~ Normal?

Answer 17

Because the normal distribution is preserved under shifting and scaling.