Problem # 1. Functions of random variables: = ae a) Let random variable X have probability density function fx(x) u(·) is the unit step function, and a > 0. Define derived random variable u(x), where Y = g(X) = loge [1 − e¯ªX] – Find the PDF ƒy(y). b) Next, consider any continuous random variable W with known PDF, fw(w). Define derived random variable Z=h(W) = log. [Fw(W)] where Fw() is the CDF of W. Find the PDF ƒz(z).

Exercise 6. Using the MGF given by Equation (24), deduce the expressions for the mean and variance we found earlier. Exercise 7. Prove the algebraic identity: 2g2 + xt = + ut 25 2g2

6. Roughly, speaking, we can use probability density functions to model the likelihood of an event occurring. Formally, a probability density function on (-x, o0) is a function f such that f(r) 20 and (2) = = 1. (a) Determine which of the following functions are probability density functions on the (-x0, 00). fr-1 00 (b) We can also use probability density functions to find the erpected value of the outcomes of the event - if we repeated a probability experiment many times, the expected value will equal the average of the outcomes of the experiment. (e.g. rf(x) dr yields the expected value for a density f(r) with domain on the real numbers.) Find the expected value for one of the valid probability densities above.

Probability and Stats