Expectation and variance using indicator random variables

Question

A group of 20 people consisting of 10 men and 10 women is randomly arranged into 10 pairs of 2 each. Compute the expectation and variance of the number of pairs that consist of two people of the opposite sex. [Hint: Use indicator random variables.]

Answer 1

Guidelines:

For $i=1,\dots,10$ let $X_i$ take value $1$ if pair $i$ consists of two people of opposite sex, and let it take value $0$ otherwise.

Then you are looking for the expectation and variance of: $$X=X_1+\cdots+X_{10}$$

Use linearity of expectation and symmetry.

Further be aware that: $$\mathsf{Var}(X)=\mathsf{Cov}(\sum_{i=1}^{10}X_i,\sum_{j=1}^{10}X_j)=\sum_{i=1}^{10}\sum_{j=1}^{10}\mathsf{Cov}(X_i,X_j)=90\mathsf{Cov}(X_1,X_2)+10\mathsf{Cov}(X_1,X_1)$$