4. Let X have pdf f (x) as follows f(x) = 1.8 1(0 1/21+ .21(1/2 < x 1) x (8 pts) What is E[X]? Justify your answ...
4. Let X have pdf (a) as follows 1) 1/2) + .2 1(1/2 < x f(z) = 1.81(0 x (8 pts) What is E[Xj? Justify your answer. (12 pts) Write Matlab or pseudocode that will produce 10, 000 PRNG samples with the dis- tribution of X. Have the output recorded in the vector a (i.e., so that (1) is the first sample, (2) is the second sample, etc.) 4. Let X have pdf (a) as follows 1) 1/2) + .2...
4. Let X have pdf f(c) as follows . (8 pts) What is EX]? Justify your answer (12 pts) Write Matlab or pscudocode that will tribution of X. Have the output recorded in the vector (i.e., so that a z(2) is the second sample, etc.) produce 10, 000 PRNG samples with the dis-
(25 pts.) Let the random variable X have pdf f(x) = { 0<x<1 1<isa Generate a random variable from f(x) using (a) The inverse-transform method (b) The accept-reject method, using the proposal density 9(x) = 0sos
4. (30 pts) Let (X,Y) have joint pdf given by e-y, 0 < x < y < 0, f(x,y) = { | 0, 0.w., (a) Find the correlation coefficient px,y. (20 pts) (b) Are X and Y independent? Explain why. (10 pts)
4. (30 pts) Let (X,Y) have joint pdf given by < , | e-9, 0 < x < f(x,y) = 3 | 0, 0.w., (a) Find the correlation coefficient px,y: (20 pts) (b) Are X and Y independent? Explain why. (10 pts)
Let X and Y have joint pdf f(x, y)= e if 0 < x < y< o and zero otherwise. Find Е(X |у). 16.
Let (X,Y) have joint pdf given by sey, 0 < x < y < 0, f(x, y) = { ( 0, 0.W., (a) Find the correlation coefficient px,y: (20 pts) (b) Are X and Y independent? Explain why. (10 pts)
Question.1 (11 Marks] Let X be a random variable with the following pdf: f(x; 8) 1>0, >0. 1 (1) (a) Show that I f(x;0)d.r = 1 (b) is the pdf f(c;) member of the Exponential family distributions? Justify in details your answer (c) Find a sufficient statistic for the unknown parameter 8. (d) Find a maximum likelihood estimator for 6.
Let the random variable X have the pdf f(x) = e^(-x) for 0 < x < 1, and 0 elsewhere. Compute the probability that the random interval (X; 3X) includes the point x = 3. What is the expected value of the length of the interval?
(Mathematical Statistics) Problem 5. Let we have a sample of size n from the pdf f(x|0) ex1(0 g(0) x < 1), e E (0, oo). Find the MLE estimator for the estimand cos(0) Problem 5. Let we have a sample of size n from the pdf f(x|0) ex1(0 g(0) x