Question

1. The number of emails that I get in one hour has such density function:
x 2 1 4 5 3 6 and more 0
f(x) 0.17 0.23 0.08 0.04 0.14 0.02 ?
a) find the probability, that there will be no letter for the next hour;
b) find cumulative distributional function.

Answer 1

$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c:c:c} x & 2 & 1 & 4 & 5 & 3 & 6\text{ and more} & 0 \\ \hline f(x) & 0.17 & 0.23 & 0.08 & 0.04 & 0.14 & 0.02 & ? \\ \end{array}$

a) $P(X=0)$ - the probability, that there will be no letter for the next hour.

To find $P(X=0)$ we can use the fact that $\sum P(X=x) =1$ .

We have: $0.17+0.23+0.08+0.04+0.14+0.02+P(X=0)=1$

$P( X=0)= 0.32$

Answer: 0.32

b) $F(x)=P(X\leq x)=P(X=0)+P(X=1)+…+P(X=x)$

Using this formula, we have:

$\def\arraystretch{1.5} \begin{array}{c:c:c:c:c:c:c:c} x & 0 & 1 & 2 & 3 & 4 &5& 6\text{ and more} \\ \hline F(x) & 0.32 & 0.55 & 0.72 & 0.86 & 0.94 & 0.98 & 1 \\ \end{array}$