Why does your title mention the moment generating function?
Hey guys, need help with this question.
Random Variables V and U are independent with N~(0,4).
Let W = U^2 + V^2 . Calculate P(W>24).
Is the Degree of Freedom here 2 ?
Why does your title mention the moment generating function?
I don't have emotions and sometimes that makes me very sad.
It happens to be at the end of my lecture notes. More specifically x^2 distribution
First, I would ask you to explain how to arrived at 2 for the degrees of freedom.
What needs to be done is to find the distribution of W, which you can ALSO do with moment generating functions, or by using relations (which can also be derived from moment generating functions). On this matter you must first do something to see what the distributions of (or rather functions of) and are.
“When things get too complicated, it sometimes makes sense to stop and wonder: Have I asked the right question?” ― Enrico Bombieri
Since they are independent, they follow a x^2 (n) distribution. How do i find out what is n ?
and have to be normalized first
So we take, and , from here you square both (which gives what distribution for them respectively, with how many respective degrees of freedom) and then you can add them together to give you .
The probability in the question must be slightly rewritten in order to use above results.
“When things get too complicated, it sometimes makes sense to stop and wonder: Have I asked the right question?” ― Enrico Bombieri
I still don't know what to do. Moreover, i don't know how you found the Mean to be 8 with Variance 64 =/
Since you want to calculate , you need to know the distribution of .
If you can only use a chi-square table, or you need to express the above probability in terms of a chi-square random variable, then you probably need to use the following facts to exploit the relationships:
1. If , then .
2. If , then
3. If and are independent, then
If I read your statement as a direct implication, then I'd say it's incorrect.Both have distributions of N[0,4] so can i say P(W>6)
The square of a standard normal random variable is a chi-squared RV with 1 degree of freedom - thats why we first normalize U & V.
Now, you can add the two of them : to obtain yet another chi-squared random variable with 2 (1 + 1) degrees of freedom - this results you can easily prove with MGF.
This leads to
From which , where
“When things get too complicated, it sometimes makes sense to stop and wonder: Have I asked the right question?” ― Enrico Bombieri
Oh, sorry about the double post above...I see BGM answered, as I was still typing.
“When things get too complicated, it sometimes makes sense to stop and wonder: Have I asked the right question?” ― Enrico Bombieri
It's obvious that the distribution is exponentional with parameter theta=8. Use this distribution to compute the probability W>24 = 0.950213.
@Dragan : Hi Dragan, I would just like to know where the "obvious" part comes into play? A more direct route is always welcome.
“When things get too complicated, it sometimes makes sense to stop and wonder: Have I asked the right question?” ― Enrico Bombieri
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