Let ? and ? be two independent uniform random variables such that ?∼????(0,1) and ?∼????(0,1). A)...

Let ? and ? be two independent uniform random variables such that ?∼????(0,1) and ?∼????(0,1).

A) Using the convolution formula, find the pdf ??(?) of the random variable ?=?+?, and graph it.

B) What is the moment generating function of ??

Solutions

Expert Solution

convolution formulae for Z=X+Y is given by: ( where X and Y are 2 continuous random variable)

Range of Z will be from 0 to 2

Now let us apply this general formula to our particular case. We will have for z<0 and also for z≥2. Now we deal with the interval from 0 to 2. It is useful to break this down into two cases (i) and (ii) 1<z<2

(i) The product is 1 in some places, and 0 elsewhere. We want to make sure we avoid calling it 1 when it is 0. In order to have, we need z−x≥0, that is, x<=z. So for (i), we will be integrating from x=0 to x=z. we have

(ii) Suppose that 1<z<2. In order to have to be 1, we need z−x<=1 that is, we need x>=z-1 So for (ii) we integrate from z−1 to 1. We have,

Graph :

milcah answered 8 months ago

Next > < Previous

Related Solutions

Problem 1 Let X1, X2, . . . , Xn be independent Uniform(0,1) random variables. (...

Problem 1 Let X1, X2, . . . , Xn be independent Uniform(0,1) random variables. ( a) Compute the cdf of Y := min(X1, . . . , Xn). (b) Use (a) to compute the pdf of Y . (c) Find E(Y ).

let x and y be independent uniform random variables over (0,1). find and sketch the pdf...

let x and y be independent uniform random variables over (0,1). find and sketch the pdf of Z=XY.

Let X,,X, and X, be independent uniform random variables on [0,1] Write Y = X, +X,...

Let X,,X, and X, be independent uniform random variables on [0,1] Write Y = X, +X, and Z = X+ X. a.) Compute E[X,X,X,. (5 points) b.) Compute Var(X). (5 points) c.) Compute and draw a graph of the density function fr. (15 points)

Suppose A and B are independent uniform random variables on [0,1]. What is the joint PDF...

Suppose A and B are independent uniform random variables on [0,1]. What is the joint PDF of (R,S)? Prove that R and S are are independent standard normal random variables. R = sqrt(-2logA)*cos(2piB) S = sqrt(-2logA)*sin(2piB)

Suppose A and B are independent uniform random variables on [0,1]. What is the joint PDF...

Given two independent uniform random variables U1 and U2 both with parameters [0,1], 1) Calculate the...

Given two independent uniform random variables U1 and U2 both with parameters [0,1], 1) Calculate the probability that 0.3 <= X + Y <= 0.8 2) Calculate the PDF of X+Y 3) Calculate the PDF of X-Y

6.42 Let X1,..., Xn be an i.i.d. sequence of Uniform (0,1) random variables. Let M =...

6.42 Let X1,..., Xn be an i.i.d. sequence of Uniform (0,1) random variables. Let M = max(X1,...,Xn). (a) Find the density function of M. (b) Find E[M] and V[M].

Let X and Y be independent Gaussian(0,1) random variables. Define the random variables R and Θ,...

Let X and Y be independent Gaussian(0,1) random variables. Define the random variables R and Θ, by R2=X2+Y2,Θ = tan−1(Y/X).You can think of X and Y as the real and the imaginary part of a signal. Similarly, R2 is its power, Θ is the phase, and R is the magnitude of that signal. (b) Find the probability density functions of R and Θ. Are R and Θ independent random variables?

(5 marks) Given two independent uniform random variables U1 and U2 both with parameters [0,1], 1)...

Given two independent uniform random variables U1 and U2 both with parameters [0,1], 1) Calculate the PDF of U1+U2 2) Calculate the PDF of U1-U2

Let ?1 and ?2 be two independent random variables with uniform distribution on [0, 1]. 1....

Let ?1 and ?2 be two independent random variables with uniform distribution on [0, 1]. 1. Write down the joint cumulative distribution function and joint probability density function of ?1 + ?2 and ?1?2

Subjects