Converting Normal Distribution to Standard Normal

[Samwise_geegee]

Homework Statement

Let X1,X2,X3 be a random sample from a normal distribution with mean μ≠0 and variance σ²=1/24. What are the values of a and b, respectively, in order for L=aX1+4X2+bX3 to have standard normal distribution?

Homework Equations

σ=1/√24

Converting normal distribution to Standard Z=(X-μ)/σ

The Attempt at a Solution

I tried to set this up by calculating Z for each value of X and setting it equal to L

aX1+4X2+bX3= (X1-μ)/(1/√24) + (X2-μ)/(1/√24) + (X3-μ)/(1/√24)

But doesn't this just mean that a=b=4? I don't really know how to tackle this problem. Any help is appreciated!

 

[Ray Vickson]

Homework Statement

Let X1,X2,X3 be a random sample from a normal distribution with mean μ≠0 and variance σ²=1/24. What are the values of a and b, respectively, in order for L=aX1+4X2+bX3 to have standard normal distribution?

Homework Equations

σ=1/√24

Converting normal distribution to Standard Z=(X-μ)/σ

The Attempt at a Solution

I tried to set this up by calculating Z for each value of X and setting it equal to L

aX1+4X2+bX3= (X1-μ)/(1/√24) + (X2-μ)/(1/√24) + (X3-μ)/(1/√24)

But doesn't this just mean that a=b=4? I don't really know how to tackle this problem. Any help is appreciated!

What are the mean and variance of L, in terms of 'a' and 'b'?