Probability if 0 For Formula 5-9, what does P(0) represent? Simplify Formula 5-9 for the case in which x = 0.

Refer to Formula 5-9, the Poisson probability of a certain event occurring x times in the defined interval is given as follows:

\(P\left( x \right) = \frac{{{\mu ^x}{e^{ - \mu }}}}{{x!}}\;\;x = 0,1,2.......\)

Let x follow a Poisson distribution with mean equal to\({\kern 1pt} \mu \).

The probability value P(0) means that the number of occurrences of the event x is equal to 0 or the event has not occurred at all.

For example, let x denote the number of trains that arrive in a day at a junction. Here, x follows a Poisson distribution with mean equal to 7 trains/day. Now, P(0) will represent the probability that no train arrived at the junction on a day.

The Poisson probability formula is as follows:

\(P\left( x \right) = \frac{{{\mu ^x}{e^{ - \mu }}}}{{x!}}\;\;x = 0,1,2.......\)

Now substituting x equal to 0, the following value is obtained:

\(\begin{aligned}{c}P\left( 0 \right) = \frac{{{\mu ^0}{e^{ - \mu }}}}{{0!}}\\ = {e^{ - \mu }}\end{aligned}\)

Where \(\mu \) is the mean number of occurrences of an event in the given interval and e is the constant with a value equal to 2.71828.