6. A random variable Y has density function fy(a)Ky(where y 2 2 (and zero otherwise) and...
Suppose the random variable X has probability density function (pdf) - { -1 < x<1 otherwise C fx (x) C0 : where c is a constant. (a) Show that c = 1/7; (b) Graph fx (х); (c) Given that all of the moments exist, why are all the odd moments of X zero? (d) What is the median of the distribution of X? (e) Find E (X2) and hence var X; (f) Let X1, fx (x) What is the limiting...
Assume that the density function for a continuous random variable, Y, is defined as fY(y) = 9y. exp(-3y) for (y>0) and f'(y) = 0 elsewhere. Given Y = y, the conditional C.D.F. for X is FX\Y (x\Y) =P[X 5 X Y = y) = 1 – exp(-x •y) for (x > 0). Questions below are related to the marginal distribution for X. 1. Derive the density, f* (x). 2. Evaluate the expectation, E(X)
Exercise 6.55 Let X and Y be random variables with joint density function f(x, y)- 4 0 otherwise Show that the joint density function of U = 3(X-Y) and V = Y is otherwise, where A is a region of the (u, v) plane to be determined. Deduce that U has the bilateral exponential distribution with density function fu (11) te-lul foru R. Exercise 6.55 Let X and Y be random variables with joint density function f(x, y)- 4 0...
If X ~U(-2; 4), find the probability density function of the random variable: a) Y = 2X + 3. b) Y = 1/(X+2)4 c) Y = X2u(X), where u(x) is the unit step function. Hint: first sketch g(x) = x2u(x)
7. (12pts.) Determine the density function for the random variable Z = X + Y where X and Y are statistically independent random variables with density functions ſi osy<1 fx(x) = 1 -21, -1<x< 1 and fy(y) = 0 otherwise »-<
1. Consider a continuous random variable X with the probability density function Sx(x) = 3<x<7, zero elsewhere. a) Find the value of C that makes fx(x) a valid probability density function. b) Find the cumulative distribution function of X, Fx(x). "Hint”: To double-check your answer: should be Fx(3)=0, Fx(7)=1. 1. con (continued) Consider Y=g(x)- 20 100 X 2 + Find the support (the range of possible values) of the probability distribution of Y. d) Use part (b) and the c.d.f....
A random variable X has probability density function given by... Using the transformation theorem, find the density function for the random variable Y = X^2 A random variable X has probability density function given by 5e-5z if x > 0 f (x) = otherwise. Using the transformation theorem, find the density function for the random variable Y = X².
2.5.6. The probability density function of a random variable X is given by f(x) 0, otherwise. (a) Find c (b) Find the distribution function Fx) (c) Compute P(l <X<3)
1. Consider two random variables X and Y with joint density function f(x, y)-(12xy(1-y) 0<x<1,0<p<1 otherwise 0 Find the probability density function for UXY2. (Choose a suitable dummy transformation V) 2. Suppose X and Y are two continuous random variables with joint density 0<x<I, 0 < y < 1 otherwise (a) Find the joint density of U X2 and V XY. Be sure to determine and sketch the support of (U.V). (b) Find the marginal density of U. (c) Find...
A random variable Y has a density fx(x) (a) What is the third quartile of X? -G #10. 0 otherwise (b) What is Fx(4)? lo