Translate and Scale Normal Distribution

Question

Working through some problems in Introduction to Probability, Blitzstein.

Let Z ~ N(0,1). Create an r.v. Y ~ N(1,4), as a simple-looking function of Z. Make sure to check that your Y has the correct mean and variance.

Definition of standardization of Normal Function is:

• $\frac{X-\mu}{\sigma}$~N(0,1)

Z~N(0,1)

Therefore:

• $\frac{Y-\mu}{\sigma}$ ~ Z
• $\frac{Y-1}{2}$ ~ Z
• Y~$2Z+1$

Is that all I'd need to do?

Answer 1

You also need to verify:

Make sure to check that your Y has the correct mean and variance.

For this, compute mean and variance of $Y$ from given mean and variance of $Z$ using

1. properties of mean and variance (e.g. linearity in case of mean. What happens in case of variance?).

2. Relation you have proposed above.

and verify that indeed you get $1$ and $4$.