Expected value of birthday problem (combination)

Question

Suppose that a child desires $10$ different toys for her birthday.

Twenty people will come to her birthday party, each of them equally likely to bring any one of the $10$ toys.

Let $X$ be the number of different types of toys brought to the party. Note that $X$ can be any integer from $1$ to $10$. What is $E[X]$?

I am asked to calculate $E[x]$.

I am basically computing $$E[X]E[X] = (1)P(1 \text{ toy type}) + (2)P(2 \text{ toy types}) + (3)P(3 \text{ toy types}) + (4)P(4 \text{ toy types}) + (5)P(5 \text{ toy types}) + \ldots$$ all the way to $10$ toy types.

I don't know if its right. Please give me a short/brief solution that I can expand on myself.

Answer 1

A short/brief hint that you can expand on yourself:

The easiest way to do this is to calculate the probability a particular toy is or is not brought and then use linearity of expectation

Answer 2

Let $X_i$ be random variable which is $1$ if child gets present $i$ and $0$ if not.

Then $X= X_1+...+X_{10}$ and $$E(X_i) = P(X_i=1) = 1-\Big({9\over 10}\Big)^{20}$$

So $$E(X) = E(X_1)+...+E(X_{10}) = 10\Big[ 1-\Big({9\over 10}\Big)^{20}\Big]$$