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1. (a) Use the definition...

1. (a) Use the definition of moment generating functions to show that if MX(t) is the mgf of a continuous random variable X, then if Y = (X+a)/b, MY (t) = e^(a/b)t MX(t/b)

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(b) Choose appropriate constants a and b, and use the result in part (a) to show that if X ∼ Normal(μ,σ2), then Z = (X−μ)/σ ∼ Normal(0,1). (You can use the mgf of a Normal random variable X as a given.)

2. Suppose that during a thunderstorm, thunderclaps occur according to a Poisson Process with rate parameter 5/hour.

(a) What is the probability that there are ten thunderclaps in the first two hours?

(b) Starting at the beginning of the storm, what is the average time until the first thunderclap is heard?

(c) What is the probability that the second thunderclap occurs after one hour has passed?

3. Let Z ∼ Normal(0, 1). Using the density of Z , show that

(a) E(Z)=0

(b) E(Z^2) = 1, and hence, Var(Z) = 1.

4. Suppose the probability density of X is given by f(x) = x/2 for 0 < x < 2 0 elsewhere.

a) Find the probability Density of Y= X^3

(b) Show that P (X ∈ (a, b)) = P (Y ∈ (a^3, b^3)) for any (a, b) ⊆ (0, 2). (Note: This illustrates the fact that, just like in the discrete case, when we transform from X to Y , the probabilities remain the same, but they pertain to different values (intervals of values) of the respective random variables.)

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