Question

Suppose that thenrandom variablesX1. . . , Xnform arandom sample from a discrete distribution for which thep.f. is f. Determine the value of Pr(X1 = X2 = . . .= Xn).

Short answer

The value of \(pr\left( {{x_1}, \ldots ,{x_n}} \right)\) is \({p^{\sum\limits_{i = 1}^n {{x_i}} }}{(1 - p)^{n - \sum\limits_{i = 1}^n {{x_i}} }}\).

Step-by-step solution

Given Information

Here given distribution is discrete distribution for \(n\) random variables.

State the random variables

For \(n\) random variables we take let \({x_1}, \ldots ,{x_n} \sim Ber\left( p \right)\) . Here we assume that the random variables follow Bernoulli distribution.

Step 3:Compute the pmf value

For discrete distribution to compute the pmf is

\(\begin{align}pr\left( {{x_1}, \ldots ,{x_n}} \right)\\ &= f\left( {{x_1}, \ldots ,{x_n}} \right)\\ &= {p^{{x_1} + , \ldots , + {x_n}}}{\left( {1 - p} \right)^{n - \left( {{x_1} + , \ldots , + {x_n}} \right)}}\\ &= {p^{\sum\limits_{i = 1}^n {{x_i}} }}{(1 - p)^{n - \sum\limits_{i = 1}^n {{x_i}} }}\end{align}\)

Hence determine the value is\({p^{\sum\limits_{i = 1}^n {{x_i}} }}{(1 - p)^{n - \sum\limits_{i = 1}^n {{x_i}} }}\).