Let $p \in (0,1)$ and $n \in \mathbb{N}$. We consider a sample of $n$ i.i.d. Bernoulli variables $X_1,\dots,X_n$ with parameter p.
Computer $E[e^{\lambda\bar{X_n}}]$ such that $\bar{X_n}= \frac{1}{n} \sum_{i=1}^n X_i$
$E[e^{\lambda\bar{X_n}}]=E[e^{\frac{\lambda}{n} \sum_{i=1}^n X_i}]=e^{\frac{1}{n}}E[e^{\lambda\sum_{i=1}^n X_i}]= e^{\frac{1}{n}}E[e^{\lambda X_1}]\dots E[e^{\lambda X_n}]=e^{\frac{1}{n}}(1-p+pe^{\lambda})^n$
Is it correct ?
$\endgroup$ 51 Answer
$\begingroup$I will use $t$ as the variable instead of $\lambda$. The required expectation is $$\left(E(\exp(\frac{tX_1}{n})\right)^n,$$ and $$E(\exp(\frac{tX_1}{n}))=(1-p)+pe^{\frac{t}{n}}.$$
$\endgroup$ 4
