The computer that controls a bank's automatic teller machine crashes a mean of 0.4 times per day. What is the probability that, in any seven-day week, the computer will crash less than 2 times? Round your answer to four decimal places.

#### Solution

Let $X$ denote the number of times a bank's automatic teller machine craches. The mean number of times a bank's automatic teller machine craches is 0.4 times per day, i.e., $E(X)= 0.4$.

In a seven-day times the $X$ follows Poisson distribution with parameter $\lambda = 7*0.4 = 2.8$.

The random variable $X\sim P(2.8)$.

The probability mass function of Poisson distribution with $\lambda =2.8$ is

` $$ \begin{aligned} P(X=x) &= \frac{e^{-2.8}(2.8)^x}{x!},\\ &\quad x=0,1,2,\cdots \end{aligned} $$ `

The probability that, in any seven-day week, the computer will crash less than 2 times is

` $$ \begin{aligned} P(X< 2) &=P(X\leq 1)\\ &= \sum_{x=0}^{1}P(X=x)\\ &=\big(P(X=0)+P(X=1)\big)\\ &= \frac{e^{-2.8}2.8^{0}}{0!}+\frac{e^{-2.8}2.8^{1}}{1!}\\ &= 0.0608+0.1703\\ &= 0.2311 \end{aligned} $$ `

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators