last day (15 days later) » 

Feeds
Feeds
04:29
1
A: Evaluating the mean and variance of the square of a normal distribution

Siong Thye GohNot quite right. You did not address the crossed term. $Y=(X_1+2X_2+X_3)^2=X_1^2+4X_2^2+X_3^2+4X_1X_2+2X_1X_3+4X_2X_3$ Hence,by independenc and linearity of expectation $E[Y] = E[X_1^2]+4E[X_2^2]+E[X_3^2]+4E[X_1]E[X_2]+2E[X_1]E[X_3]+4E[X_2]E[X_3]$ Now, you can evaluate the individual terms. Now,...

spiiju
spiiju
don't you think cross-terms be zero as x1,x2,x3 are independent?
Siong Thye Goh
Siong Thye Goh
No. Independence just means $E[X_1X_2]=E[X_1]E[X_2]$.
spiiju
spiiju
got it, thanks. and to evaluate E(X1^2) I have to use variance formula, right?
Siong Thye Goh
Siong Thye Goh
yup. $E[X_1^2]=Var(X_1)+E[X_1]^2$.
spiiju
spiiju
I got E[Y] = 77 using the first approach. Could you please tell me how to use the second moment of the normal distribution to get this?
Siong Thye Goh
Siong Thye Goh
04:29
$E[Y]=E([(X_1+2X_2+X_3)^2]$ this is equal to the second moment of $X_1+2X_2+X_3$ which is just equal to $2^2+8^2+3^2=77$ since the expectation of $X_1+2X_2+X_3$ is $0$.
spiiju
spiiju
Thank you for this approach. One more doubt to compute 𝜎̃ ^4 I have to 2^4+8^4+3^4 or 77^2?
Siong Thye Goh
Siong Thye Goh
hi, since $\tilde{\sigma}^2 = 77$, we have $\tilde{\sigma}^4 = 77^2$.
in general, square of the sum is not equal to the sum of square of each term
just starting the chat to avoid long comment threads
spiiju
spiiju
Hello
so, i got E[Y^2] =12579
Siong Thye Goh
Siong Thye Goh
how do you obtain that number?
spiiju
spiiju
3(2^4+8^4+3^4) = 12579
Siong Thye Goh
Siong Thye Goh
04:35
we can't just power 4 each individual term
spiiju
spiiju
So, how to compute E[Y^2] using the 4th moment of norma distribution?
Siong Thye Goh
Siong Thye Goh
the formula is $3\tilde{\sigma}^4$
the formula for the moments can be found on the wikipedia page
spiiju
spiiju
how to compute sigma^4?
Siong Thye Goh
Siong Thye Goh
$77^2$
spiiju
spiiju
so, E[Y^2] will be 17787
3 * 77*77
Siong Thye Goh
Siong Thye Goh
04:41
yup
spiiju
spiiju
variance is coming 11858 ( 17787 - 77^2)
Siong Thye Goh
Siong Thye Goh
seems to be so
spiiju
spiiju
Thank you so much, Siong
Siong Thye Goh
Siong Thye Goh
welcome. remember for future post, always include your attempt and use descriptive title. :)
spiiju
spiiju
yes, I'll. I'm new to stackexchange still figuring out things but will surely do from next time
Siong Thye Goh
Siong Thye Goh
04:47
welcoem to the site, don't be discouraged from the negative votes :) see you around the site sometimes
spiiju
spiiju
Thank you so much
one more question, can you tell what distribution will be E[Y] having?
normal distribution, right?
 
11 hours later…
Siong Thye Goh
Siong Thye Goh
16:02
E[Y] is a number

last day (15 days later) »