Question

Suppose that X1 and X2 are independent standard normal random variables. Show that Z = X1...

Suppose that X1 and X2 are independent standard normal random variables. Show that Z = X1 + X2 is a normal random variable with mean 0 and variance 2.

Math   Statistics-And-Probability   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

Given:

X1 is Standard Normal Variate. So, Probability Density Function of X1 is given by:

X2 is Standard Normal Variate. So, Probability Density Function of X2 is given by:

The Probability Density Function of

Z = X1 + X2 is given by Convolution Theorem as follows:

The expression in the brackets = 1, since it is the integral of the Normal Density Function with = 0 and = .

So,

we have:

This proves the required result that Z = X1 + X2 is a normal random variable with mean 0 and variance 2.

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
Suppose X1 and X2 are independent expon(λ) random variables. Let Y = min(X1, X2) and Z...
Suppose X1 and X2 are independent expon(λ) random variables. Let Y = min(X1, X2) and Z = max(X1, X2). (a) Show that Y ∼ expon(2λ) (b) Find E(Y ) and E(Z). (c) Find the conditional density fZ|Y (z|y). (d) FindP(Z>2Y).
If X1 and X2 ~ Normal(0,1) are independent standard normally distributed random variables, Calculate 1) cov(X1,...
If X1 and X2 ~ Normal(0,1) are independent standard normally distributed random variables, Calculate 1) cov(X1, X2) 2) cov(3X1, X2 /3) 3) cov(X1, 0.8X2 + ((1-0.8^2)^0.5) X2) 4) cov(X1, -0.4X2 + ((1-0.4^2)^0.5) X2)
Suppose X1, X2, X3, and X4 are independent and identically distributed random variables with mean 10...
Suppose X1, X2, X3, and X4 are independent and identically distributed random variables with mean 10 and variance 16. in addition, Suppose that Y1, Y2, Y3, Y4, and Y5are independent and identically distributed random variables with mean 15 and variance 25. Suppose further that X1, X2, X3, and X4 and Y1, Y2, Y3, Y4, and Y5are independent. Find Cov[bar{X} + bar{Y} + 10, 2bar{X} - bar{Y}], where bar{X} is the sample mean of X1, X2, X3, and X4 and bar{Y}...
Let X1, X2,... be a sequence of independent random variables distributed exponentially with mean 1. Suppose...
Let X1, X2,... be a sequence of independent random variables distributed exponentially with mean 1. Suppose that N is a random variable, independent of the Xi-s, that has a Poisson distribution with mean λ > 0. What is the expected value of X1 + X2 +···+ XN2? (A) N2 (B) λ + λ2 (C) λ2 (D) 1/λ2
Let Y be the liner combination of the independent random variables X1 and X2 where Y...
Let Y be the liner combination of the independent random variables X1 and X2 where Y = X1 -2X2 suppose X1 is normally distributed with mean 1 and standard devation 2 also suppose the X2 is normally distributed with mean 0 also standard devation 1 find P(Y>=1) ?
Suppose that X1, X2, . . . , Xn are independent identically distributed random variables with...
Suppose that X1, X2, . . . , Xn are independent identically distributed random variables with variance σ2. Let Y1 = X2 +X3 , Y2 = X1 +X3 and Y3 = X1 + X2. Find the following : (in terms of σ2) (a) Var(Y1) (b) cov(Y1 , Y2 ) (c) cov(X1 , Y1 ) (d) Var[(Y1 + Y2 + Y3)/2]
Suppose that X1 and X2 are independent continuous random variables with the same probability density function...
Suppose that X1 and X2 are independent continuous random variables with the same probability density function as: f(x) = ( x 2 0 < x < 2, 0 otherwise. Let a new random variable be Y = min(X1, X2,). a) Use distribution function method to find the probability density function of Y, fY (y). b) Compute P(Y > 1).
The random variables X1 and X2 both follow normal distributions. The mean of X1 is E(X1)=5,...
The random variables X1 and X2 both follow normal distributions. The mean of X1 is E(X1)=5, and its variance is V(X1)=2 The mean of X2 is E(X2)=9, and its variance is V(X2)=3. If Y is a random variable such that Y = 3X1+5X2, what is P(Y<70)? A student takes 4 measurements and finds that the mean is 64 and the sample variance is 81. What is the sample standard deviation For a random variable X, which statement is most likely...
Suppose that X1 and X2 are independent continuous random variables with the same probability density function...
Suppose that X1 and X2 are independent continuous random variables with the same probability density function as: f(x) = ( x 2 0 < x < 2, 0 otherwise. Let a new random variable be Y = min(X1, X2,). a) Use distribution function method to find the probability density function of Y, fY (y). b) Compute P(Y > 1). c) Compute E(Y )
1)Let X1, ..., Xn be independent standard normal random variables, we know that X2 1 +...
1)Let X1, ..., Xn be independent standard normal random variables, we know that X2 1 + ... + X2 n follows the chi-squared distribution of n degrees of freedom. Find the third moment of the the chi-squared distribution of 2 degrees of freedom. 2) Suppose that, on average, 1 person in 1000 makes a numerical error in preparing his or her income tax return. If 10,000 returns are selected at random and examined, find the probability that 6 or 7...
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT