91Ó°ÊÓ

X is a random variable such that\(E\left( {{x^k}} \right) = \frac{1}{3}\)for k=1,2,3...

Since\(E\left( {{x^k}} \right)\)has the same value for every positive integer we will find a random variable X such that\(X,{X^2},{X^3},{X^4}....\)all have the same distribution.

If X takes values 0 and 1

Then\({X^k} = X\)for every positive integer k

Since\({0^k} = 0\)and\({1^k} = 1\).

if

\(\begin{aligned}{c}{\rm P}\left( {X = 1} \right) = p\\ = 1 - {\rm P}\left( {X = 0} \right)\end{aligned}\)

then

\(E\left( {{X^k}} \right) = \frac{1}{3}\).

As required ,we must have\(p = \frac{1}{3}\)

Therefore The random variable x such that\({\rm P}\left( {X = 1} \right) = \frac{1}{3}\)and\({\rm P}\left( {X = 0} \right) = \frac{2}{3}\)

Have Bernoulli distribution with parameter\(\frac{1}{3}\)