(1) Derive and plot the probability density function of Z.
(2) Plot the cumulative distribution function of Z.
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(1) Derive and plot the probability density function of Z. (2) Plot the cumulative distribution f...
PART E ONLY 2. (25 pts) Determine the expectation, variance, standard deviation, and cumulative distribution function...
PART E ONLY
2. (25 pts) Determine the expectation, variance, standard deviation, and cumulative distribution function (c.d.f.) for each of the following: f(x)0.44 0.36 0.15 0.04 0.01 (b) f(x) = 끊 for z = 1, 2, 3, 4 (c) f(x) = 5F-302145 (d) A random variable that represents the outcome of rolling one (fair) die. (e) A random variable that represents the outcome of rolling two (fair) dice for 1,2,4,5
2) Consider a random variable Z with a uniform probability density function given as UZ(-1,0) and...
2) Consider a random variable Z with a uniform probability
density function given as UZ(-1,0) and X=4Z+4. a) Find and plot the
probability density function ( ) Xf x . b) Find and plot the
probability distribution function ( ) F x X . c) Find E[Z]. d) Find
E[X]. e) Find the correlation of Z and X. i. Are they correlated?
ii. Are they independent? Why?
2) Consider a random variable Z with a uniform probability density function given...
he cumulative distribution function (cdf), F(z), of a discrete ran- om variable X with pmf f(x)...
he cumulative distribution function (cdf), F(z), of a discrete ran- om variable X with pmf f(x) is defined by F(x) P(X < x). Example: Suppose the random variable X has the following probability distribution: 123 45 fx 0.3 0.15 0.05 0.2 0.3 Find the cdf for this random variable
For a random variable X with cumulative distribution function (cdf) Fx(x) = 1- (2/x)^2 ,x>2. (a).Find...
For a random variable X with cumulative distribution function (cdf) Fx(x) = 1- (2/x)^2 ,x>2. (a).Find the pdf fX(x). (b).Consider the random variable Y = X^2. Find the pdf of Y, fY (y).
Open book, open notes. No collaboration. Return this sheet along with your answers (17) 1. Assume...
Open book, open notes. No collaboration. Return this sheet along with your answers (17) 1. Assume that a binary communication system sends message "O" as -5 V and message l" as +5 V randomly but with a "I" three times as likely as a "O". Because of uniformly-distributed noise picked up during transmission, a "o" arrives at the receiver input as a voltage uniformly distributed between -7 V and -3 V, and a "" arrives at the receiver input as...
Let X be a random variable with probability density function fx(2, -1 9. f(y) y> 9. Submit Answe...
WILL THUMBS UP IF DONE
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Let X be a random variable with probability density function fx(2, -1 <z<3, 0 otherwise. Find the probability distribution of Y-X2 for 0 < y < 1, 1 < y < 9, and y > 9. [Obviously, fy(y)-0 for y < 0.1 Case 1: O < y < 1. Enter a formula below. Use * for multiplication, / for divison, ^ for power and sqrt for square root. For example, sqrt y...
3. The cumulative distribution function of a random variable Y is: 0 if y<-1, 0.3 if...
3. The cumulative distribution function of a random variable Y is: 0 if y<-1, 0.3 if -1 y <0.5, 0.7 Fr (y)- y < 2, if 0.5 1 if y 2 2. (a) Draw a sketch plot of Fy (y) d) Find the probability mass function, fz(2), of Z -Y2 (e Find El2] and Var(Z) (f) Find El2-321 and Var(2-32). 13 marks]
2) A random variable X has the density function: fr(x) =[u(x-1)-u(x-3)]. Define event B (Xs 2.5)...
2) A random variable X has the density function: fr(x) =[u(x-1)-u(x-3)]. Define event B (Xs 2.5) (a) Find the cumulative distribution function, Fy (x). (b) Find the conditional distribution Fx (x|B). the mean E[X], and variance of X Fx(xB)= E[X)= Variance (e) Sketch both Fy(x) and Fx (x|B) on the same plot. Show all important values. (d) Let the output of random variable X above be applied to a square-law device according to Y 5X2. Find the mean value of...
1. Consider a random experiment that has as an outcome the number x. Let the associated random variable be X, with true (population) and unknown probability density function fx(x), mean ux, and varia...
1. Consider a random experiment that has as an outcome the number x. Let the associated random variable be X, with true (population) and unknown probability density function fx(x), mean ux, and variance σχ2. Assume that n 2 independent, repeated trials of the random experiment are performed, resulting in the 2-sample of numerical outcomes x] and x2. Let estimate f x of true mean ux be μΧ-(X1 + x2)/2. Then the random variable associated with estimate Axis estimator Ax- (XI...
Problem 3. Define the function: 2+_ 0 if (z,y)#10.0) if (a,y)-(0,0) f(x, v)= (a) Graph the...
Problem 3. Define the function: 2+_ 0 if (z,y)#10.0) if (a,y)-(0,0) f(x, v)= (a) Graph the top portion of the function using Geogebra. Does the function appear to be continuus at 0? (b) Find fz(z, y) and fy(z, y) when (z, y) #10.0) (c) Find f(0,0) and s,(0,0) using the limit definitions of partial derivatives and f,(0,0)-lim rah) - f(O,0) d) Use these limit definitions to show that fay(0,0)--1, while x(0,0)-1 (e) Can we conclude from Clairaut's theorem that()-yr(x,y) for...
PART E ONLY
2. (25 pts) Determine the expectation, variance, standard deviation, and cumulative distribution function (c.d.f.) for each of the following: f(x)0.44 0.36 0.15 0.04 0.01 (b) f(x) = 끊 for z = 1, 2, 3, 4 (c) f(x) = 5F-302145 (d) A random variable that represents the outcome of rolling one (fair) die. (e) A random variable that represents the outcome of rolling two (fair) dice for 1,2,4,5
2) Consider a random variable Z with a uniform probability
density function given as UZ(-1,0) and X=4Z+4. a) Find and plot the
probability density function ( ) Xf x . b) Find and plot the
probability distribution function ( ) F x X . c) Find E[Z]. d) Find
E[X]. e) Find the correlation of Z and X. i. Are they correlated?
ii. Are they independent? Why?
2) Consider a random variable Z with a uniform probability density function given...
he cumulative distribution function (cdf), F(z), of a discrete ran- om variable X with pmf f(x) is defined by F(x) P(X < x). Example: Suppose the random variable X has the following probability distribution: 123 45 fx 0.3 0.15 0.05 0.2 0.3 Find the cdf for this random variable
Open book, open notes. No collaboration. Return this sheet along with your answers (17) 1. Assume that a binary communication system sends message "O" as -5 V and message l" as +5 V randomly but with a "I" three times as likely as a "O". Because of uniformly-distributed noise picked up during transmission, a "o" arrives at the receiver input as a voltage uniformly distributed between -7 V and -3 V, and a "" arrives at the receiver input as...
WILL THUMBS UP IF DONE
NEATLY AND CORRECTLY!
Let X be a random variable with probability density function fx(2, -1 <z<3, 0 otherwise. Find the probability distribution of Y-X2 for 0 < y < 1, 1 < y < 9, and y > 9. [Obviously, fy(y)-0 for y < 0.1 Case 1: O < y < 1. Enter a formula below. Use * for multiplication, / for divison, ^ for power and sqrt for square root. For example, sqrt y...
3. The cumulative distribution function of a random variable Y is: 0 if y<-1, 0.3 if -1 y <0.5, 0.7 Fr (y)- y < 2, if 0.5 1 if y 2 2. (a) Draw a sketch plot of Fy (y) d) Find the probability mass function, fz(2), of Z -Y2 (e Find El2] and Var(Z) (f) Find El2-321 and Var(2-32). 13 marks]
2) A random variable X has the density function: fr(x) =[u(x-1)-u(x-3)]. Define event B (Xs 2.5) (a) Find the cumulative distribution function, Fy (x). (b) Find the conditional distribution Fx (x|B). the mean E[X], and variance of X Fx(xB)= E[X)= Variance (e) Sketch both Fy(x) and Fx (x|B) on the same plot. Show all important values. (d) Let the output of random variable X above be applied to a square-law device according to Y 5X2. Find the mean value of...
1. Consider a random experiment that has as an outcome the number x. Let the associated random variable be X, with true (population) and unknown probability density function fx(x), mean ux, and variance σχ2. Assume that n 2 independent, repeated trials of the random experiment are performed, resulting in the 2-sample of numerical outcomes x] and x2. Let estimate f x of true mean ux be μΧ-(X1 + x2)/2. Then the random variable associated with estimate Axis estimator Ax- (XI...
Problem 3. Define the function: 2+_ 0 if (z,y)#10.0) if (a,y)-(0,0) f(x, v)= (a) Graph the top portion of the function using Geogebra. Does the function appear to be continuus at 0? (b) Find fz(z, y) and fy(z, y) when (z, y) #10.0) (c) Find f(0,0) and s,(0,0) using the limit definitions of partial derivatives and f,(0,0)-lim rah) - f(O,0) d) Use these limit definitions to show that fay(0,0)--1, while x(0,0)-1 (e) Can we conclude from Clairaut's theorem that()-yr(x,y) for...
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