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 $$\sigma$$. Find the constants a and b so that the random variable y=aX+b has a Gaussian PDF with mead m' and standard deviation $$\sigma$$'.

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 $$aX + b$$ to have a different mean but same standard deviation? I'm not entirely clear from your post.

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

If $$\mu_1$$ is supposed to be the new mean, then

$$E(aX+b) = aE(X) + b = \mu_1$$

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

$$Var(aX+b) = \sigma^2$$

Simplifying and working with these equations will let you find appropriate values for $$a, b$$. Play with them.

 

[ahamdiheme]

no the new deviation is $$\sigma'$$

 

[statdad]

I'm not sure what you mean by saying "the new standard deviation is $$\sigma'$$

Is it simply that

$$\sigma' = \sqrt{Var(aX+b)}$$

 

[ahamdiheme]

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

that relationship, i know. the m' goes with $$\sigma'$$.
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