Mean and variance of a distribution

You will write functions for the mean and the variance of a distribution stored as a dictionary, and for the distribution of $aX + b$. The checker uses the heads in three tosses of a coin.

The idea. A distribution is a dictionary that maps each value $x$ to its probability. The mean is $\sum x\,p$. The variance is $E(X^2) - [E(X)]^2$. The variable $aX + b$ takes the value $ax + b$ with the same probability as $x$.

Your tasks.

1. Finish mean(pmf). Add x * p for every value.
2. Finish variance(pmf). Find the mean of the squares and subtract the square of the mean.
3. Finish shift_scale(pmf, a, b). Return a new dictionary with the keys a * x + b.