. Let W and V be independent random variables where W has a normal distribution with...
Exercise 6 Let Yi, Y2, Ys be independent random variables with distribution N (i, i2) for i = 1, 2, 3 (that is, each is normally distributed with mean mean E(Y) = i and variance V(X) = i2). For each of the following situations, use the Y, i = 1, 2, 3 to construct a statistic with the indicated distribution a) X2 with 3 degrees of freedom b) t distribution with 2 degrees of freedom c) F distribution with 1...
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.
(Sums of normal random variables) Let X be independent random variables where XN N(2,5) and Y ~ N(5,9) (we use the notation N (?, ?. ) ). Let W 3X-2Y + 1. (a) Compute E(W) and Var(W) (b) It is known that the sum of independent normal distributions is n Compute P(W 6)
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...
Let Xi, x,, ,X, be independent random variables with mean and variance σ . Let Y1-Y2, , Y, be independent random variables with mhean μ and variance a) Compute the expected value of W b) For what value of a is the variance of W a minimum? σ: Let W-aX + (1-a) Y, where 0 < a < 1. Let Xi, x,, ,X, be independent random variables with mean and variance σ . Let Y1-Y2, , Y, be independent random...
Let X and Y be independent random variables. Random variable X has a discrete uniform distribution over the set {1, 3} and Y has a discrete uniform distribution over the set {1, 2, 3}. Let V = X + Y and W = X − Y . (a) Find the PMFs for V and W. (b) Find mV and (c) Find E[V |W >0].
Let X1, ..., Xn be a random sample from a Normal distribution with mean zero and standard deviation sigma. Let X bar and S^2 be the sample mean and sample variance, respectively. a. Find the constant c such that c(Xbar2) / S^2 has an F distribution. b. How many degrees of freedom are associated with this F distribution?
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...
Exercise 8.43. Let Z1, Z2,... . Zn be independent normal random variables with mean 0 and variance 1. Let (a) Using that Y is the sum of independent random variables, compute both the mean and variance of Y. (b) Find the moment generating function of Y and use it to compute the mean and variance of Y. Exercise 8.43. Let Z1, Z2,... . Zn be independent normal random variables with mean 0 and variance 1. Let (a) Using that Y...
Exercise 6.15. Let Z, W be independent standard normal random variables and-1 < ρ < l. Check that if X-Z and Y-p2+ VI-p-W then the pair (X, Y) has standard bivariate normal distribution with parameter ρ. 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.