QUESTION 4 Let X, Y be joint Gaussian random variables with E[X]=0,E[Y]=0, Var[X]=2, Var[Y] =4 and Cov[X, Y] =2. Compute E[XIY=1] as a decimal and enter the answer.
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- Question 3 A product is classified according to the number of defects it contains and the factory that produces it. Let X and Y be the random variables that represent the number of defects per unit (taking on possible values of 0, 1, 2, or 3) and the factory number (taking on possible values 1 or 2), respectively. The entries in the table represent the joint possibility mass function of a randomly chosen product.Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0X is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2
- Suppose X and Y are independent. X has a mean of 1 and variance of 1, Y has a mean of 0, and variance of 2. Let S=X+Y, calculate E(S) and Var(S). Let Z=2Y^2+1/2 X+1 calculate E(Z). Hint: for any random variable X, we have Var(X)=E(X-E(X))^2=E(X^2 )-(E(X))^2, you may want to find EY^2 with this. Calculate cov(S,X). Hint: similarly, we have cov(Z,X)=E(ZX)-E(Z)E(X), Calculate cov(Z,X). Are Z and X independent? Are Z and Y independent? Why? What about mean independence?Let Xi be IID random variables which have the same law as X. Let L(t) = E(e^tX.) Suppose that this is well defined for t ∈ [−1, 1]. Express the moment generating function of the Sum from i=1 to k Xi in terms of k and LConsider the following Gauss/Jordan reaction:
- If we let RX(t) = ln MX(t), show that R X(0) = μ and RX(0) = σ2. Also, use these results to find the mean and the variance of a random variable X having the moment-generating function MX(t) = e4(et−1)Let X and Y be random variables, and a and b be constants. a) Prove that Cov(aX, bY) = ab Cov(X,Y). b) Prove that if a > 0 and b > 0, then ρaX,bY = ρX,Y. Conclude that the correlation coefficient is unaffected by changes in units.Answer the following questions. Let X be a continuous random variable with P(X<0)=0. When E(X)=\mu exists, P(X\ge 3\mu) \le \frac{1}{(a)} by the Markov's inequality. What is (a)? Consider two random variables X and Z. The relationship between X and Z is given as X=1+2Z. Let Z be a random variable with moment generating function (mgf), M_Z(t) = (1-t)^{-3}, for t<1. What is the expectation of X?