Conditional distribution

Conditional distribution of Y given X = x, P(Y = y | X = x) isans.

conditional probability, independence of events

For each of the following pairs of random variables X and Y , the conditional distribution of Ygiven {X = x} is a named distribution, with parameters that may depend on x. For each pair:1i. Write down the conditional distribution of Y given {X = x}, including parameter(s).(2 marks each)ii. Find E (Y | X). (1 mark each)iii. Find E(Y ). (1 mark each)iv. Find Cov(X, Y ). (2 marks each)You do not need to find the marginal distributions of X and Y , which might not be nameddistributions.(a) Choose a number between 10 and 29, uniformly at random, and call that number X.Take a standard deck of 52 cards and lay them out in a row, face up. Using a pen, puta checkmark on the first X cards. Then turn the cards face down, shuffle throughly, anddraw 5 cards. Let Y be the number of marked cards you draw.

What are conditionals

Continuing from the previous problem, suppose X has c.d.f. given by F(x) = 1-e^(-sqrt(x)), x>0. a) Show that Y=sqrt(X) follows the exponential distribution with parameter 1. b) Using the previous result, compute the conditional probability P(X>=9|X>=4). The p.d.f. of an exponential randome variable with parameter λ is f(x) = λe^(-λx), for x>=0.