# hw5 - X F X x(c The median of a distribution of a...

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STAT 225 - Homework 5 Due Thursday June 4 Do the following problems. 1. Let Y Poisson ( λ ). (a) Show that the moment generating function of X is: M Y ( t ) = e λ ( e t - 1) . Guidance: You’ll have to use the following math fact: X k =0 a k k ! = e a (b) Diﬀerentiate the mgf in (a) to ﬁnd E ( Y ) and E ( Y 2 ). Then ﬁnd V ( Y ). 2. Problem 1.147 on page 142 gives you the moment generating function of the geometric random variable. Use it to do problem 3.148. 3. Read on page 141 starting with “The second (but primary). ..” through example 3.25 and then do: Page 142, problem 3.149 Page 142, problem 3.150. 4. Page 167, problem 4.8 (a), (b) and (e). 5. Page 167, problem 4.12 (c), (d) and (e) Note: for parts (c) and (d) remember that Y measures time in hundreds of hours . 6. Verify that the following function is a legitimate density function. f X ( x ) = 0 . 2 e - 2 x + 0 . 9 e - x x > 0 0 otherwise 7. Let X have the density function given by: f X ( x ) = x 2 9 0 x c 0 otherwise (a) Find c so that f X ( x ) is a legitimate density function. (b) Find the cdf of

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Unformatted text preview: X , F X ( x ). (c) The median of a distribution of a continuous random variable X (denoted by m ) is the value that half the area under the density function is to the left of it and half the area is to the right. In other words, the median m satisﬁes: P ( X ≤ m ) = . 5, which means that the random variable X is as likely to get values below m as it is to get values above m . Find the median of the distribution above. 1 8. Page 173, problem 4.30 (a) and (b) only 9. Page 173, problem 4.31 10. A family of pdf’s that has been used to approximate the distribution of income, city popula-tion size, and size of ﬁrms is the Pareto family. The family has two parameters, k and θ , and the pdf is: f X ( x ) = kθ k x k +1 x ≥ θ otherwise If k > 1, compute E ( X ) and explain why you need the fact that k > 1. 2...
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hw5 - X F X x(c The median of a distribution of a...

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