Question

Let X be a Bernoulli random variable (Hint: Special case of Binomial Distribution)

a. Compute E(X²)

b. Show that V(X) is p(1-p)

c. Compute E(X⁷⁹)

Answer 1

Solution:

$X \sim \operatorname{Bernoulli}(p), 0 \leq p \leq 1$

So, $P(X=x)= \begin{cases}1-p & \text { if } x=0 \\ p & \text { if } x=1\end{cases}$  

Now the expected value of X:

 $\begin{aligned} E(X) &=0 \cdot P(X=0)+1 \cdot P(X=1) \\ &=0 \times(1-p)+1 \times p \\ &=p \end{aligned}$  

(a):

 $\begin{aligned} E\left(X^{2}\right) &=0^{2} \cdot P(X=0)+1^{2} \cdot P(X=1) \\ &=0 \times(1-p)+1 \times p \\ &=p \end{aligned}$  

(b):

 $\begin{aligned} \operatorname{Var}(X) &=E\left(X^{2}\right)-E^{2}(X) \\ &=p-p^{2} \\ &=p(1-p) \end{aligned}$

(c):

$\begin{aligned} E\left(X^{79}\right) &=0^{79} \cdot P(X=0)+1^{79} \cdot P(X=1) \\ &=0 \times(1-p)+1 \times p \\ &=p \end{aligned}$