Given the joint pdf S4x²y+ 2y5; 0

Two random variables X and Y have joint char- acteristic function (0,,m)=exp(-2@²– 8m3) (i) Show that X and Y are zero mean ran- dom variables.

2. Suppose that X1, . Xn are iid Geometric random variables with frequency func- ... tion f(x; 0) = 0(1 – 0)", x = 0, 1, 2, ..., 0 E (0, 1). Find the ML estimator 0, of 0. Show that Ôn is consistent an find its asymptotic distribution.

b) A continuous random variable X has the p.d.f f(x) = {A(2 – x)(2 + x), 0 < x < 2, Find (i) the value of A, (ii)P(X <1)(iii) P(1 < X <2). l0, otherwis ------------

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Which of the following is false for continuous and independent random variables Y and Z?

Let X; (i = 1, 2,..., n) be i.i.d. exponential random variables with E{X;} = 1/A. Let Y = mn{X1, X2,..., Xn}. What is the distribution of Y? (Compute P{Y > y}.) What is the distribution of Z = max{X1, X2, ..., Xn}? (Compute P{Z < z}.)

Let X1,X2.,X, and Y1,Y2,.Yn be two random sample of size n from a normal independent distributions Var(X)=1 and Var(Y)=20?. Let U = E-1(X; – X)² and W = E1(Y; – Y)? W a. What is the sampling distribution of the statistic U + 202 b. If the 90th percentile of the statistic (U + 202 is 33.20, what is the sample size n?

Let X be a random variable corresponding to 2021 oil production in New Mexico relative to 2020.  Let Y be a random variable corresponding to 2021 oil production in Texas relative to 2020.   Suppose the joint distribution of X and Y is    f (x,y) = 4xy   for x,y SET [0,1] Find P (X < 1/2 | Y > 3/4)          ** X is less than or equal to 1/2  and Y is greater than or equal to 3/4

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