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Suppose three coins are tossed and we are interested to determine the
number of heads that will come out. Let us use H to represent the number
of heads that will come out. List the sample space and count the number of
heads in each outcome and assign this number to this outcome.
2. A company holds a raffle game. The amount to be won are as follows: one Php10, 000; two Php5, 000 each and four Php2, 000 each. Two thousand tickets were sold at Php100 each. If you buy one of the raffle tickets, what is your expected net gain?
Let X1, X2 have the joint probability density function f(x1, x2) = 2e −(x1+x2) , 0 < x1, x2 < ∞ Let Y1 = X1, Y2 = X2 − X1.
(i) Using the change of variable technique, find the joint probability density function of Y1, Y2
(ii) Find the conditional distribution of Y2 given Y1
The joint probability density function of two random variables X1 and X2 is defined by f(x1, x2, x3) = 2, 0 < x1 < x2 < 1
Find the conditional distribution of X1 given X2 = x
The moment generating function of two jointly distributed random variables X1 and X2 is M(t1, t2) = e ^− 0.5 G where G = (7.51t 2 1 + 7.9t 2 2 + 3.8574t1t2 + 135.4t1 + 137.2t2) Using this function, find the correlation coefficient of of X1 and X2
Let X and Y have joint probability distribution function f(x, y) = ( 2x+y /12) , (x, y) = (0, 1); (0, 2); (1, 2); (1, 3) 0, elsewhere
Find
(i) the covariance between X and Y.
(ii) the joint probability generating function of X and Y.
Let X and Y be two independent random variables having joint probability density function f(x, y) = 1/ 2πσ2 e − (x−µ) 2 σ2 e − (y−µ) 2 σ2 − ∞ < x, y < ∞
Find the moment generating function of Z = X+Y 2 and hence the mean and variance of Z
A discrete random variable X has probability distribution function f(x) = 12! /x!(12−x)!p x (1 − p) 12−x x = 0, 1, 2, .., 12 0, elsewhere
(i) if p = 0.3, find Pr(X > 3).
(ii) find possible values of p if Var[X] is equal to 1.92.
Let A and B be two events defined on a sample space S. If Pr(A)=0.8; Pr(A| B )=0.85 and Pr(A| B^ c )=0.75; determine the probability that neither of the two events occur.
The scores of individual students on a national test have a normal distribution with mean 18.6 and standard deviation 5.9. At Bagabag National High School, 76 students took the test. If the scores at this school have the same distribution as national scores, what are the mean and standard deviation of the sample mean for 76 students?
