Suppose we have a distribution that has some pdf, say, $f(x)= 4(3-x^3)$ for $0<x<2$, and $0$ elsewhere. I am able to find $E(X)$ and $Var(X)$ using the standard definition of expected value (i.e. integral of $xf(x)$) and variance. But suppose we are given that a random sample of some size, say $15$, has been obtained from this distribution, and so this sample itself will have a mean, and the problem is to find the expected value of the sample mean. I initially thought that the expected value and variance of the sample mean will just be $E(X)$ and $Var(X)$ respectively, but I am not sure if the sample mean and variance will be influenced by the sample size.
My query is whether the expected value of the sample mean is the same as $E(X)$ regardless of sample size, and if not what is the approach to determining the expected value of the sample mean taking into account the sample size.