1. Let X be an RV with PDF a. Find the mean of X. b. Find the variance of X. f(x) = {¹ -=| - |x| for x < 1 otherwise

Suppose X and Y are two independent variables with variance 1. Let Z = X+bY where b > 0. If Cor(Z, Y ) = 1/2, what is the value of b?

Let X1 and X2 be IID exponential with parameter  > 0. Determine the distribution ofY = X1=(X1 + X2).

Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.

x > 1 A RV X has the PDF fx(x) = otherwise (a) Find the MGF Mx(t) of X and clearly define the values of t for which it exists. (b) Find the mean (µ) and variance (o?) of the RV X using the MGF from (a).

TF.5 A small hotel has 5 rooms. Let X be the number of occupied rooms on any given day. The pmf p(x) of X is given in the table. x 0 1 2 3 4 5 p(x) .05 .05 .20 .25 .30 .15 Find the means E(X) and E(X2) , the variance V(X) and the standard deviation SD(X) of X.

Pv. Don't provide handwriting solution

Suppose that n observations are chosen at random from a continuous pdf fY(y).  What is the probability that the last observation recorded will be the smallest number in the sample?

3. 1 + x Check whether the following function is a valid pdf? If it is valid, obtain its CDF. If not valid, would scaling by a constant make it valid? If so, make it valid. Obtain its CDF? What is its mean? Variance? ƒ = —exp(-√x)U(x)

* Let E(X]Y = y) = 3y and var(X|Y = y) = 2, and let Y have the p. d. f. se-y if y > 0 fG) = {o otherwise Then the variance of X is