4. Let X, Y, and Z be independent random variables, each with the standard normal distribution....
4. Let X, Y, and Z be independent random variables, each with the standard normal distribution. Compute the following: (a) P[X + Y> Z +2 (b) Var3x 4Y;
xercise 6.15. Let Z, W be independent standard normal random variables and-1 < ρ < 1 . Check that if X Z and Y-: ρΖ+ VI-P" W then the pair (X, Y) has standard bivariate normal distribution with parameter p. Hint. You can use Fact 6.41 or arrange the calculation so that a change of variable in the inner integral of a double integral leads to the right density function.
1. Let Xi, X2,... be independent random variables each with the standard normal distribution, and for each n 2 0 let Sn-1 Xi. Use importance sampling to obtain good estimates for each of the following probabilities: (a) Pfmaxn<100 Sn> 10; and (b) Pímaxns100 Sn > 30) HINTS: The basic identity of importance sampling implies that d.P n100 where Po is the probability measure under which the random variables Xi, X2,... are independent normals with mean 0 amd variance 1. The...
2. If X and Y are independent random variables, X has a normal distribution with mean 2 variance 4, and Y has a chi-square distribution with 9 degrees of freedom, then find u such that P(X > 2+11,7)=0.01.
Suppose that the standard normal random variables X and Y are independent. Find P(0 < X<Y). 8 O 1 4T 0 1 8л Ala
number? 10 3. Let X be a continuous random variable with a standard normal distribution. a. Verify that P(-2 < X < 2) > 0.75. b. Compute E(지)· 110]
1. Let X1, X2,... be independent random variables each with the standard normal distribution, and for each n 0 let Sn 너 1 i. Use importance sampling to obtain good estimates for each of the following probabilities: (a) P[maxns 100 Sn > 10); and (b) P[maxns100 Sn > 30 HINTS: The basic identity of importance sampling implies that n100 where Po is the probability measure under which the random variables Xi, X2,... are independent normals with mean 0 amd variance...
4.3. Let X and Y be independent random variables uniformly distributed over the interval [θ-, θ + ] for some fixed θ. Show that W X-Y has a distribution that is independent of θ with density function for lwl > 1.
Let X, Y be two independent exponential random variables with means 1 and 3, respectively. Find P(X> Y)
Let the random variable Z follow a standard normal distribution. Find P(-2.35 < Z< -0.65). Your Answer: