1
$\begingroup$

We have been discussing pmf. Can someone please show me

A. That the pmf for a binomial random variable sums to 1

B. That the pmf for a geometric random variable sum to 1

So I may see the difference

  • 0
    What do you mean by A and$1$b? For any discrete random variable, the sum of the pmf over all possible values is $1$. This is because the total probability (i.e. the probability of the whole sample space) is always $1$.2012-09-30

1 Answers 1

3

A. You have to show $\sum_{i=0}^n {n \choose i}p^i(1-p)^{n-i}=1$ but the left hand side is $(p+(1-p))^n=1.$

B. Depending on your definition, you have to show $\sum_{i=0}^\infty p(1-p)^{i}=1 \text{ or } \sum_{j=1}^\infty p(1-p)^{j-1}=1 $ and they are clearly the same sums if you replace $i$ by $j-1$.

The geometric series $\sum_{i=0}^\infty x^i = \frac{1}{1-x}$ for $|x|\lt 1$, so (taking the former) $p\times \sum_{i=0}^\infty (1-p)^{i}=p\times \dfrac{1}{1-(1-p)}=1.$