QUESTION 4 The bivariate beta type Il distribution has the probability density function a-1,b-1 x...
Suppose density function positively valued continuous random variable X has the probability a fx(x)kexp 20 fixed 0> 0 for 0 o0, some k > 0 and for (a) Find k such that f(x) satisfies the conditions for a probability density function (4 marks) (b) Derive expressions for E[X] and Var[X (c) Express the cumulative distribution function Fx(r) in terms of P(), the stan dard Normal cumulative distribution function (8 marks) (8 marks) (al) Derive the probability density function of Y...
Recall that if X has a beta(a, B) distribution, then the probability density function (pdf) of X is where α > 0 and β > 0. In this problem, we are going to consider the beta subfamily where α-β θ. Let X1, X2, , Xn denote an iid sample from a beta(8,9) distribution. (b) The two-dimensional statistic nm 27 is also a sufficient statistic for θ. What must be true about the conditional distribution (c) Show that T* (X) is...
4. Use the distribution function technique to find the density function for Y = 2X + 3 The density function for X is f(x). Your answer should be given as a piecewise function. 2x + 1) 1<x<2 f(x) = 4 0 elsewhere =f2x+1) h 5. Use the transformation technique to find the density function for Y = 4x + 1. The density function for X is f(x). Your answer should be a piecewise function. f(x) = S4e-4x 0 < x...
Question 1(a&b) Question 3 (a,b,c,d) QUESTION 1 (15 MARKS) Let X and Y be continuous random variables with joint probability density function 6e.de +3,, х, у z 0 otherwise f(x, y 0 Determine whether or not X and Y are independent. (9 marks) a) b) Find P(x> Y). Show how you get the limits for X and Y (6 marks) QUESTION 3 (19 MARKS) Let f(x, x.) = 2x, , o x, sk: O a) Find k xsl and f(x,...
QUESTION6 (a) The three-parameter gamma distribution has the probability density function x (r)- exp (r-c)-1,x> Derive the mean of the distribution. (b) Ifx beta I (m, n) show thatY ax +b(l-X) has the four-parameter beta distribution with parameters a, b, m and n.
1. Consider the joint probability density function 0<x<y, 0<y<1, fx.x(x, y) = 0, otherwise. (a) Find the marginal probability density function of Y and identify its distribution. (5 marks (b) Find the conditional probability density function of X given Y=y and hence find the mean and variance of X conditional on Y=y. [7 marks] (c) Use iterated expectation to find the expected value of X [5 marks (d) Use E(XY) and var(XY) from (b) above to find the variance of...
< 1. The joint probability density function (pdf) of X and Y is given by for(x, y) = 4 (1 - x)e”, 0 < x <1, 0 < (a) Find the constant A. (b) Find the marginal pdfs of X and Y. (c) Find E(X) and E(Y). (d) Find E(XY).
1. The joint probability density function (pdf) of X and Y is given by fxy(x, y) = A (1 – xey, 0<x<1,0 < y < 0 (a) Find the constant A. (b) Find the marginal pdfs of X and Y. (c) Find E(X) and E(Y). (d) Find E(XY). 2. Let X denote the number of times (1, 2, or 3 times) a certain machine malfunctions on any given day. Let Y denote the number of times (1, 2, or 3...
The random variable X has probability density function f (x) = k(−x²+5x−4) 1 ≤ x ≤ 4 or =0 1 Show that k = 2/9 Find 2 E(X), 3 the mode of X, 4 the cumulative distribution function F(X) for all x. 5 Evaluate P(X ≤ 2.5). 6 Deduce the value of the median and comment on the shape of the distribution.
The following joint probability distribution is given. 1. Find k such that the given function demonstrates the PDF. 2. Find Marginal distributions. 3. Evaluate ?(? < ? < 0) 4. Find the correlation coefficient between X and Y having the joint density functions:(.) ?(?,?) = {???2+?2 ??? ?2 + ?2 < 4 0 ?????h??? Question 2. (20 pts.) The following joint probability distribution is given. 1. Find k such that the given function demonstrates the PDF. 2. Find Marginal distributions....