Finding Constants for a Gaussian PDF

[ahamdiheme]

Homework Statement

The exam grades in a certain class have a Gaussian PDF with mean m and standard deviation [tex]\sigma[/tex]. Find the constants a and b so that the random variable y=aX+b has a Gaussian PDF with mead m' and standard deviation [tex]\sigma[/tex]'.

Homework Equations

The Attempt at a Solution

I really do not know where to go from here, i need a heads-up.
Thanks

 

[statdad]

Is [tex] aX + b [/tex] to have a different mean but same standard deviation? I'm not entirely clear from your post.

You do know that if [tex] X [/tex] is Gaussian then [tex] aX + b [/tex] is also Guassian for any choices of [tex] a \ne 0 \text{ and real }b [/tex], right, so you don't need to show that part.

If [tex] \mu_1 [/tex] is supposed to be the new mean, then

[tex]
E(aX+b) = aE(X) + b = \mu_1
[/tex]

The other condition requires you to work with the variances: If the standard deviation doesn't change then you know that

[tex]
Var(aX+b) = \sigma^2
[/tex]

Simplifying and working with these equations will let you find appropriate values for [tex] a, b [/tex]. Play with them.

 

[ahamdiheme]

no the new deviation is [tex]\sigma'[/tex]

 

[statdad]

I'm not sure what you mean by saying "the new standard deviation is [tex] \sigma' [/tex]

Is it simply that

[tex]
\sigma' = \sqrt{Var(aX+b)}
[/tex]

 

[ahamdiheme]

y=aX+b has a Gaussian PDF with mean m' and standard deviation '

that relationship, i know. the m' goes with [tex]\sigma'[/tex].
Hope u understand what the question says now. It seems a little confusing but that's the exact way the textbook put it. Thank you