How to find density function?

Question

$X \sim N(1,4)$ and $Y = 3 - 5X$. How to find the density function of $Y$?

I tried first to find the distribution function of Y, but got stuck.

$$F(y) = P(Y <= y) = P(3 - 5X <= y) = P(X >= (-y + 3)/5).$$

Any help appreciated.

score 2 · Accepted Answer · answered May 22 '13 at 14:36

2

Hint: Try to calculate the mean and variance of $Y$. $Y$ also has a normal density, just its mean and variance are different from $X$.

answered May 22 '13 at 14:36

Matt L.

score 0 · Answer 2 · answered May 22 '13 at 14:40

Note that: $$P(X >= (-y + 3)/5)= 1 - P(X <= (-y + 3)/5) = 1 - \Phi((-y+3)/5,1,4)$$

where,

$\Phi(x,\mu,\sigma)$ stands for the cdf of a normal with mean $\mu$ and standard deviation $\sigma$.

Now differentiate both sides to get the pdf of $Y$ and show that it is a normal with the appropriate mean and variance.

2 Answers2