I am struggling with the two questions of this exercise :
Let $N \in \mathbb{N}^*$ and $ [\![ 1, N ]\!] $ the set of natural numbers $k$ such that : $1 \leq k \leq N$. A roulette allows to make the equiprobable drawing of the elements of $ [\![ 1, N ]\!] $.
$1)$ Let $p$ be a natural number. $p$ consecutive draws are made using the roulette. Let $X$ be the random variable which indicates the cardinal of the set obtained. Calculate the expected value of $X$
$2)$The drawings of the roulette are repeated until all the whole numbers from $1$ to $N$ have been drawn. Let $Y$ ne the random variable indicating the rank of the stroke. Calculate the expected value of $Y$
I must say that I have completely no ideas of how to proceed for question $2$. Yet for question $1$, I noticed that :
$$P(X = m) = \frac{\binom {N} {m} \cdot (m)^{p-m}}{N^p}$$ Hence if I am not mistaken by the definition of the expected value, we have : $$E(X) = \displaystyle\sum_{i = 1}^{p} p(X = i) \cdot i$$ Yet as you can see It's seems impossible to get a close form ...