can someone of you say me if the following statement is true?
For $i = 1,...,n$ let $X_i$ be a real-valued random variable with density function $f_i$ on $(\mathbb{R}, B(\mathbb{R}))$.
Then the function $f(x) :=\frac{f_1(x) + ... + f_n(x)}{n}$, $x ∈ \mathbb{R}$, is the density of a random variable.
For me the statement is true since $\int_{-\infty}^{\infty}f(x)dx=\int_{-\infty}^{\infty}\frac{f_1(x) + ... + f_n(x)}{n}dx=\int_{-\infty}^{\infty}\frac{f_1(x)}{n}+...+\frac{f_n(x)}{n}dx=n\frac1 n$=1