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- 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?Show that variance o? = (x²) – ((x))²Prove that the variance of b, V(b) = o²(X'X)−¹
- 4. Suppose that X has pdf f(x) = 3x² for 0 < x< 1. Find the pdf of the random variable Y = VX.Suppose X and Y are two random variables with E[X] = 1, Var (X) = 4, E[Y ] = -1, Var (Y) 4, and Cov (X, Y) = 1. Find the standard deviation of (X - Y). = (a) 2 (b) √2 (c) 6 (d) √6 (e) None of the above4. Let X be a positive random variable (i.e. P(X 1/E(X) (b) E(-log(X)) > -log(E(X)) (c) E(log(1/X))> log(1/E(X)) (d) E(X³) > (E(X))³
- Let E(Y|X = x) = 2x and var(Y|X = x) = 4x², and let X have p. d. f. f(x) = {',;" s1,0 < x <1 0, e.w. Then the variance of Y is8. Given the Beta Distribution where p(x) = B(a,b).ªªa- Calculate the mean and variance B(a, b) = -xª-¹ (1-x)b-1 1 = √₁² 2²- x-¹(1-x)-¹dx = r(a) I (b) I(a + b)If the moment-generating function of X is 2 M(t) = e' + find the mean, variance, and pmf of X. 2t
- Suppose that Y₁ = 0.5, Y₂ = 0.2, Y4 = 0.7 and Y5 = 0.6, represents a random sample. Each of these Y's comes from the same population and has as a density of fy, (vi) = (0+1)yi; 0 -1 It can be shown that the natural logarithm of the likelihood function is equal to: n* ln(0 + 1) + 0 *Σln(yi) a. Determine the form of the maximum likelihood estimator for 0. b. Use the MLE formula and the data provided to find an estimate for 0. = 0.6, Y3Suppose X1, X2,..., Xn are i.i.d. random variables with PDF p(x) = ¿e-¤/® for x > 0 and 0 is a positive constant. Show that X is an unbaised estimator for 0. Find MSE(X) in estimating 0. Show that X² is a baised estimator for 0² . Find MSE(X²) in estimating 0². Suggest an unbaised estimator for 0²3. Let X1, and E- X; are independent. , Xn be iid with exponential distribution with mean 0. Show that (X1+X2)/ E=1 X;