Ind rand variable distribution

In summary, the conversation discusses the distribution of a variable Z that is the sum of two independent normal random variables X and Y. It is mentioned that the characteristic function of Z is the product of the characteristic functions of X and Y, and that this leads to Z being normally distributed with a variance of 2. The last question involves setting up a ratio of double integrals and converting to polar coordinates to solve for the distribution. The final answer is not explicitly stated, but it is mentioned that Z is normally distributed with a mean of 0 and a variance of 2.
  • #1
johnny872005
12
0
Hi Guys.

I'm trying to understand how to solve the following problem. Any help and explanation would be greatly appreciated!
thanks.

Let X and Y be independent N(0, 1) random variables and let Z = X + Y.

What is the distribution of Z? Write down the density function of Z.
Also:
Show that E[Z|X > 0, Y > 0] = 2
 
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  • #2
Have you learned about characteristic functions(cf)? If so then you get the cf of Z is the product of the cf's of X and Y. The cf of Z is then exp(-t2), so that Z is normally distributed with a variance of 2.

For the last question, set it up as a ratio of double integrals in x and y. Then convert to polar coordinates. It should be tedious, but workable.
 
  • #3
If X~N(m1,s1^2) and Y~N(m2,s2^2) indeply, then
X(+-)Y ~ N(m1(+-)m2,s1^2+s2^2), it is easier to check from mgf or cf.
 
  • #4
I'm still somewhat confused. What would you say as the final answer to the two questions I have above...?

If you could just list something like:
answer: thefunc
reason: because x y and z

Sorry if it seems bossy, I'm just really confused now after reading the above.
thanks
 
  • #5
mathman said:
so that Z is normally distributed with a variance of 2.


It's hardly possible to state the answer to 1) more explicitly.
 
  • #6
Pere Callahan said:
It's hardly possible to state the answer to 1) more explicitly.

No, you have to add 'and mean=0'. LOL.
 
  • #7
O.k., you're right, of course.:smile:
 

1. What is an independent random variable?

An independent random variable is a variable that is not affected by any other variable in the system. This means that the value of the variable is not influenced by any other factors and is completely random. It is often used in statistical analysis to represent a variable that is not dependent on any other factors.

2. How is an independent random variable different from a dependent random variable?

An independent random variable is not affected by any other variables, while a dependent random variable is influenced by at least one other variable. This means that the value of a dependent random variable is not completely random, but is instead influenced by other factors in the system.

3. What is the probability distribution of an independent random variable?

The probability distribution of an independent random variable is a function that assigns probabilities to all possible values that the variable can take on. This function is often represented graphically as a histogram or a probability density function, and it shows how likely it is for the variable to take on each possible value.

4. How does the distribution of an independent random variable affect its behavior?

The distribution of an independent random variable can greatly affect its behavior. For example, a variable with a normal distribution will behave differently than one with a uniform distribution. The distribution can also affect the likelihood of certain values occurring and can provide insight into the behavior of the variable in a system.

5. How are independent random variables used in scientific research?

Independent random variables are commonly used in scientific research for statistical analysis and modeling. They can represent variables that are not influenced by other factors, making them useful for studying cause and effect relationships. They are also used in simulations and experiments to represent the randomness inherent in many natural systems.

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