# Question: Let f x 1 1 x2 x

Let f(x) = 1/[π(1 + x2)], −∞ < x < ∞, be the pdf of the Cauchy random variable X. Show that E(X) does not exist.

## Answer to relevant Questions

Let X be N(0, 1). Find the pdf of Y = |X|, a distribution that is often called the half-normal. Hint: Here y ∈ S y = {y : 0 < y < ∞}. Consider the two transformations x1 = −y, −∞ < x1 < 0, and x2 = y, 0 < y < ∞. When α and β are integers and 0 < p < 1, we Have Where n = α + β − 1. Verify this formula when α = 4 and β = 3. Let X have a beta distribution with parameters α and β. (a) Show that the mean and variance of X are, respectively (b) Show that when α > 1 and β > 1, the mode is at x = (α − 1)/(α + β − 2). Let X1 and X2 be a random sample of size n = 2 from the exponential distribution with pdf f(x) = 2e−2x, 0 < x < ∞. Find (a) P(0.5 < X1 < 1.0, 0.7 < X2 < 1.2). (b) E[X1(X2 − 0.5)2]. Let X1 and X2 be two independent random variables. Let X1 and Y = X1 + X2 be χ2(r1) and χ2(r), respectively, where r1 < r. (a) Find the mgf of X2. (b) What is its distribution?Post your question