Question

A branch of a certain bank in New York City has six ATMs. Let $X$ represent the number of machines in use at a particular time of day. The cdf of $X$ is as follows:
$f\left(x\right)=\left\{\begin{array}{l}\begin{array}{c}0x<1\\ .601\le x<1\\ \begin{array}{c}.191\le x<2\\ .392\le x<3\\ \begin{array}{c}.673\le x<4\\ .924\le x<5\\ .975\le x<6\end{array}\end{array}\end{array}\\ 16\le x\end{array}\right.$
Calculate the following probabilities directly from the cdf:
$P\left(2\right),$that is,$P(X=2)$

Answer 1

Finding the probabilities of $P(X=2)$ from the given function:

Given cumulative distribution function is:

$f\left(x\right)=\left\{\begin{array}{l}\begin{array}{c}0x<1\\ .601\le x<1\\ \begin{array}{c}.191\le x<2\\ .392\le x<3\\ \begin{array}{c}.673\le x<4\\ .924\le x<5\\ .975\le x<6\end{array}\end{array}\end{array}\\ 16\le x\end{array}\right.$

$P(X=2)$ means the value of function at $X=2$,
From above function we can see $P(X=2)=0.39(2\le x<3)$

Hence, the answer is $0.39$.