Show that V(X) = 0 if and only if there is a constant c such that P(X c) 1.
I Let fx,r(x,Y)— il(x +9)
Find V(2X — 3Y + 8).
0IA fair coin is tossed until a head is obtained. What is the expected number of tosses that will be required?
'Let X be a continuous random variable with CDF F. Suppose that P(X > 0) = 1 and that E(X) exists. Show that E(X) = P(X > x)dx. Hint: Consider integrating by parts. The following fact is helpful: if E(X) exists then xil — F(x)I = 0.
Prove Theorem 3.17. IProve the formulas given in the table at the beginning of Section 3.4 for the Bernoulli, Poisson, Uniform, Exponential, Gamma, and Beta. Here are some hints. For the mean of the Poisson, use the fact that = Er.0 ax/x!. To compute the variance, first compute E(X(X — 1)). For the mean of the Gamma, it will help to multiply and divide by r(o+I)/ir+1 and use the fact that a Gamma density integrates to 1. For the Beta, multiply and divide by Net + 1)1"(/3)/r(cr + 0 + 1). ISuppose we generate a random variable X in the following way. First we flip a fair coin. If the coin is heads, take X to have a Unif(0,1) distribution. If the coin is tails, take X to have a Unif(3,4) distribution. (a) Find the mean of X. (b) Find the standard deviation of X.
***only uniform and exponential need. dont need others
ILet the random variable X denote an individual's blood calcium level and the random variable Y denote his or her blood cholesterol level. The joint PDF is 1 fx,r(x, = {240 0 otherwise Derive the marginal PDF of X, E(X), V(X), Derive the marginal PDF of Y, E(Y), V(Y), Derive E(XY), Cov(X, Y), and interpret the covariance between X and Y.
8.5