SOLUTION:
a Find Fand Y he as defined in Problem 1. I define a new random variable...
Define a random variable , and a new random variable Y, such that 1) Find the density function of Y.( Instruction: Find the the cumulative distribution function and the derivative it) 2) Find the expectation of Y for (Hint: look for its connection with normal distribution of random variable) T~erp(A) We were unable to transcribe this imageWe were unable to transcribe this image
Let X be exponential random variable with λ = 1. (a) Define Y = √ X. Specify the support of Y and find its density. (b)Define Z = X^2 + 2X. Specify the support of Z and find its density.
Let X be a random variable with CDF z<0 G()=/2 0 <IS2 z>2 1 Suppose Y = X2 is another random variable, find (a) P(1/2 X 3/2), (b) P(1s X< 2) (c) P(Y X) (d) P(X 2Y). (f) If Z VX, find the CDF of Z. (d) P(X+Y 3/4)
2. Det X be a geometric random variable with mean S. Define a new random variable Y using the following function Y-11,-31 ifXcS 2 ifX25 Where| | denote the absolute value. (a) Find the PMF ofY (b) Find the CDF of Y (c) Find E[Y] and Var(Y] (d) Find P IYel Y 3]
2. Det X be a geometric random variable with mean S. Define a new random variable Y using the following function Y-11,-31 ifXcS 2 ifX25 Where| |...
Consider the random variable Y, whose probability density function is defined as: if 0 y1 2 y if 1 y < 2 fr(v) 0 otherwise (a) Determine the moment generating function of Y (b) Suppose the random variables X each have a continuous uniform distribution on [0,1 for i 1,2. Show that the random variable Z X1X2 has the same distribution = as the random variable Y defined above.
Consider the random variable Y, whose probability density function is defined...
Assume random variable ? is uniformly distributed in the
interval (−?/2 ,?⁄ 2]. Define the random variable ?=tan (?), where
tan (∙) denotes the tangent function. Note that the derivative of
tan (?) is 1/(cos (?)2) .
a) Find the PDF of ?.
b) Find the mean of ?
.Define the random variable ?=1/?.
c) Find the PDF of ?.
Assume random variable X is uniformly distributed in the interval (-1/2, 1/2). Define the random variable Y = tan(X), where...
Two-dimensional random variable has probability density function which is defined as f(x,y)=c(x+2y) , when 0<y<1 and 0<x<2, but 0 otherwise. Find the constant c, find the marginal density functions of X and Y and find if X and Y are independent.
We have a random variable, X. Using the variable, we construct a new variable Y, defined below: Y = 3X+5. Calculate the mean and variance of Y in terms of X. (i) E(Y) (ii) Var(Y)
4.4.19 Random variableX has PDE fx(a)-1/4 -1s-33, 0 otherwise Define the random variable Y by Y = h(X)X2. (a) Find E[X and VarX (b) Find h(E[X]) and Eh(X) (c) Find ElY and Var[Y .4.6 The cumulative distribution func- tion of random variable V is 0 Fv(v)v5)/144-5<7, v> 7. (a) What are EV) and Var(V)? (b) What is EIV? 4.5.4 Y is an exponential random variable with variance Var(Y) 25. (a) What is the PDF of Y? (b) What is EY...
(1) Let X be exponential random variable with λ = 1. (a) (4 pts) Define Y = √ X. Specify the support of Y and find its density. Show all of your work and computations. (b) (6 pts) Define Z = X^2 + 2X. Specify the support of Z and find its density. Show all of your work and computat