Question

Determine whether it can serve as the probability distribution of a random variable X. Explain your answer.

P(X)= 3+x / 3-x for X= 1, 2, 3, 4

Answer 1

We are given that,

$P(X)= (3+x) / (3-x) \space for \space x= 1, 2, 3, 4$

Now,

$P(1)={(3+1)\over(3-1)}={4\over2}=2\\ P(2)={(3+2)\over(3-2)}={5\over1}=5\\ P(3)={(3+3)\over(3-3)}={6\over0}=\infin\\ P(4)={(3+4)\over(3-4)}={7\over-1}=-7\\$

For $P(x)$ to be a probability distribution then it must satisfy the condition that $\sum P(X)=1$

So,

$\sum P(X)=2+5+\infin+-7=\infin$

Clearly, $P(X)$ cannot serve as a probability distribution because, $\displaystyle\sum_{\forall x} P(X)\not=1$