2. Let X and Y be independent integer-valued RVs with given distributions jezqj = 1. (a) Compute ...
Problem 5) Let X and Y be independent gamma RVs with parameter (a, 1) and (3, 1), respec- tively. a) Show that X + Y is also gamma RV with parameters (a +3,1). b) Compute the joint density of U = X + Y and V = ty
2. Let the joint probability density function of (X, Y) be given by {ay otherwise. 1 and 0 < y < 2, f(z,y) (a) [6 pts] Determine if X and Y are independent. (b) [6 pts] Find P{X+Y <1) B( (c) [6 pts) Find 2. Let the joint probability density function of (X, Y) be given by {ay otherwise. 1 and 0
Problem 3. Let X and Y be two independent random variables taking nonnegative integer values (a) Prove that for any nonnegative integer m 7m k=0 b) Suppose that X~ B (n, p) and Y ~ B(m. p), and X, Y are independent. What is the distribution of the random variable Z X + Y? (c) Prove the following formula for binomial coefficients: n\ _n + m for kmin (m, n) (d) Let X ~ B (n, 1/2). What is P...
Concept Check: Conditional Quantile 1 point possible (graded) Let (X, Y) be a pair of RVs with joint density f (x, y) = x + y, over the sample space 12 = [0, 1]?. For a given x, what is the value qa (x) such that P[Y < 9a (x) |X = x] = 1 – a? That is, what is the conditional (1 – a)-quantile function (of x) of Y|X = x? The Shapes of Joint and Conditional Distributions...
1. Let X and Y be two independent random variables following beta distributions Beta(120, 2019) (a) What's P(X 0.3)? (b) What's E(2X - Y)? (c) What's P(2X +4 > 3Y)? (d) What's P(X < Y)? (e) Now if X and Y are no longer independent to each other. Will the answers to a)-(d) remain the same? Explain. (f) Now define Z~Beta(2019, 120). Compare the median of X and Z, which one is bigger? Compare the variance of X and Z,...
Let X and Y be independent exponential(1) RVs (f(x) e 10). Show that uniform(0, 1) distribution. Hint: consider defining the auxiliary X/(X Y) has a RV XY [12
The joint probability density function of random variables X and Y is given by f(x,y) ={10xy^2 0≤x≤y≤1,0 otherwise. (a) Compute the conditional probability fX|Y(x|y). (b) Compute E(Y) and P(Y >1/2). (c) Let W=X/Y. Compute the density function of W. (d) Are X and Y independent? Justify briefly.
7.70. Let X,...,x,; Y,., Y,; Z,..,Z, be respective independent random samples from three normal distributions N(u,a+ B, a) N(4-B+y,a), N(= a + y , a'). Find a point estimator for B that is based on X, Y, Z. Is this estimator unique? Why? If a is unknown, explain how to find a confidence interval for B. 7.70. Let X,...,x,; Y,., Y,; Z,..,Z, be respective independent random samples from three normal distributions N(u,a+ B, a) N(4-B+y,a), N(= a + y ,...
5. Suppose that X and Y are independent with distributions N(0,0) and N(0,02), respectively. Let Z=X+Y. Also, let W = 02X – oʻY. Prove that Z and W are uncorrelated.
1. Let X and Y be two independent random variables following beta distributions Beta (120, 2019). (a) what's P(X=0.3)? (b) What's E(2X -Y)? (c) What's P(2X 4> 3Y)? d) What's P(X< Y)? (e) Now if X and Y are no longer independent to each other Will the answers to (a)-(d) remain the same? Explain (f) Now define Beta(2019,120). Compare the median of X and Z, which one is bigger Compare the variance of X and Z, which one is biggr?...