TOPIC:Transformation of random variables.
Below are sample questions: [5] 6. Let X ~ GAMMA(0,k). Prove that Y = 2x 2K)
Let X, X2, ..., X, be independent with X-Gamma (a,b). Let Y = EX. Prove that Y-Gamma (a,b)
Let X ~ Gamma(k, β) and Y ~ Gamma(k, 1) Gamma( α, 3) Cx Show that Y = 스 is a pivot Let X ~ Gamma(k, β) and Y ~ Gamma(k, 1) Gamma( α, 3) Cx Show that Y = 스 is a pivot
6. Let f(x,y) = 1 if 0 < y < 2x, 0<x<1, and 0 otherwise. Find the following: a) f(y|x) b) E(Y|X = x) c) The correlation coefficient, p, between X and Y
Prove: Let k be a positive integer, and set n :=2k-1(2k – 1). Then (2k+1 – 1)2 = 8n +1 Prove: Let n be a positive integer, and let s and t be integers. Show that Hire (st) = n(s) in (t) mod n.
5-4. Prove that the MGF of Gamma distribution is В f(t) В — t. 5-5. Let X1, X2,... ,X be independent with X~ Gamma (ai,ß). Let Y = EX4. Prove that Y Gamma (Eai,ß)
10) (11) Let X and Y be 2 independent random variables. Suppose X ~ Gamma(0, 38) and Y ~ Gamma(a, 2B). Let 2 = 2X +3Y. Determine the probability distribution of Z. (Hint: use the method of moment-generating functions
4. Let 8 >0. Let X, X2,..., X, be a random sample from the distribution with probability density function S(*;ð) - ma t?e-vor x>0, zero otherwise. Recall: W=vX has Gamma( a -6, 0-ta) distribution. Y=ZVX; = Z W; has a Gamma ( a =6n, = ta) distribution. i=1 E(Xk) - I( 2k+6) 120 ok k>-3. 42 S. A method of moments estimator of 8 is 42.n 8 = h) Suggest a confidence interval for 8 with (1 - 0) 100%...
prove that J2(x)=sum from k=0 to infinity [ (-1)^k/2^9@k+2)*k!(k+2)! ]*x^(2k+2) is a solution of the Bessel differential equation of order 2: x^2y'' + xy' + (x^2-4)y=0 (-1)4 9- Using the ratio test, one can easily show that the series +2converges for all e R. Prove that (-1)X h(x) = E, 22k +2k!(k + 2)! 22+2 is a solution of the Bessel differential equation of order 2: In(x) is called the Bessel function of the first Remark. In general the function...
Let X have a gamma distribution with parameters a > 2 and 3 > 0. (a) Prove that the mean of 1/X is B/(a - 1). (b) Prove that the variance of 1/X is 82/[(a - 1)(a 2)].
answer should be 2x 5. Let X andY joint density function if 0r< 1; 0 <y<r 8.ry f(r,y) = 0 elsewhere. What is the regression curve y on r, that is, E (Y/X = r)?