What is the mean value of $\max\limits_{j}X_j$? Let $X_j$ be a random variable that is $1$ with probability $x_j^*$, and $0$ with probability $1-x_j^*$. The random variables $X_j$ are independent and $j$ belongs to $\{1,\ldots,n\}$ for some positive integer $n$. I would like to calculate the mean value of $\max\limits_{j}X_j$. That is, 
$$\mathbb{E}\left[\max\limits_{j}X_j\right]$$
My try is: since $X_j$ is a binary random variable, than the maximum would be $1$. Therefore, 
$$\mathbb{E}\left[\max\limits_{j}X_j\right]=\Pr\left[X_j=1\right]=x_j^*,$$
but I do not know if this is the right argument.

The max is either $0$ or $1$.
The probability that it's $0$ is the probability that all (independent) $X_j$ are $0$, that is
$$P(max X_j =0)=\prod_j P(X_j=0)=\prod_j (1-x^*_j)$$
And of course, 
$$P(max X_j =1)=1-\prod_j (1-x^*_j)$$
Thus $E(max X_j)=1-\prod_j (1-x^*_j)$
