Suppose X and Y are random variables such that has a normal distribution with mean and standard deviation ? = 1.
a). (4 points) Find a formula for E[Y |X = x].
b.) Compute E[Y]
Suppose X and Y are random variables such that has a normal distribution with mean and...
Suppose X and Y are random variables such that fY (y|X = x) has a normal distribution with mean µ = x/4 and standard deviation σ = 1. a). Find a formula for E[Y|X = x]. b). Compute E[Y ].
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.
4. Suppose X and Y are standard normal random variables. Find an expression for P (X +2Y-3) in terms of the standard normal distribution function Φ in two cases: (a) X and Y are independent; (b) X and Y have bivariate normal distribution with correlation ρ = 1/2·
10. (5pt) Suppose that X and Y are two normally distributed random variables. X has mean 2 and standard deviation !5 Y has mean 5 and standard deviation 3. Their correlation is 0.6. What is the mean and standard deviation of X + Y? What is the distribution of X+ Y? What if X and Y are jointly normally distributed? What if they are not jointly normally distributed? Explain your answer.
Suppose X and Y are standard normal random variables. Find an expression for P (X + 2Y-3) in terms of the standard normal distribution function Φ in two cases: (a) X and Y are independent; (b) X and Y have bivariate normal distribution with correlation p 1/2.
Suppose Z and X are continuous random variables such that Z has a standard normal distribution and X = 5% + 10. a. Compute P(7 < X < 17). [6] b. What are the expected value E(X) and variance V(X) of X? [6] c. What kind of distribution does X have? [3]
6. (a) Given that X and Y are continuous random variables, prove from first principles that: (b) The random variable X has a gamma distribution with parameters-: 3 and A-2 . Y is a related variable with conditional mean and variance of =x)= Calculate the unconditional mean and standard deviation of Y. (c) Suppose that a random variable X has a standard normal distribution, and the conditional distribution of a Poisson random variable Y, given the value ol XOx, has...
4. Suppose X and Y are independent random variables with the same probability distribution, given by the cumulative distribution function if t 2 1 if t < 1 F(t)= 1 -t-3 (a) (10 points) Compute E(X). (b)(10 points) Compute E(XY). Chr
Suppose two continuous random variables X and Y have cumulative distribution functions Fx(x) and Fy(y) respectively. Suppose that Fx(x) > Fy(x) for all x. Indicate whether the following statements are TRUE or FALSE with brief explanation. (a) E(X) > E(Y) (b) The probability density functions fx, fy satisfy fx(x) > fy(x) for all x. (c) P (X = 1) > P (Y = 1)
Please show steps (and formulas) for part b Problem 2. a. X has a normal distribution with mean 5 and variance 25. Y has a normal distribution with mean 3 and variance 16. In addition, X and Y are independent. If W = X+Y, find P(W > 9). b. Random variables U, V, Z are such that E[U] = 1, E[V] =5, E[2] = -3, Var[U] = 1, Var[V] = 4, Var[2] =1, Cov[U,V] =-1,Cov[U, 2] = 2, Cor[V, 2]...