#### 13. Creating Formulas

*Question 1*

E(*Y*) = E(*X*_{1} + ... + *X*_{n}) = E(*X*_{1}) + ... + E(*X*_{n}) = μ + ... + μ = *n*μ.

Assuming the *X*_{j} are all independent,

var(*Y*) = var(*X*_{1}) + ... + var(*X*_{n}) = = σ^{2} + ... + σ^{2} = *n*σ^{2}.

Thus sd(*Y*) = √n σ.

*Question 2*

The distribution for *X* given by Table 11.1 is just the Binomial(3, 0.5) distribution. Using the formulas from Section 13.3 we find

E(*X*) = 3 × 0.5 = 1.5, and

sd(*X*) = √(3 × 0.5 × 0.5) = √0.75 = 0.866.

Of course these are the same as the values obtained from Table 11.1 using the original definitions of expected value and standard deviation.