Probabilities of the Form P(x)

To compute P(x), use

dpois(x, \(\mu\))

For example, suppose x = 8 and \(\mu\) = 10. Find P(8).

dpois(8,10)
## [1] 0.112599

So, P(8) = 0.1126

Probabilities of the Form P(X \(\leq\) x) or P(X < x)

To compute probabilities of the form P(X \(\leq\) x), use the command

ppois(x, \(\mu\))

For example, suppose \(\mu\) = 8, find P(X \(\leq\) 16).

ppois(16,10)
## [1] 0.9729584

So, P(X \(\leq\) 16) = 0.9730.

To compute P(X < x), you must change the probability to one of the form P(X \(\leq\) x). For example, the possible values for a Poisson random variable X are x = 0, 1, 2, 3, 4, 5, … . So, P(X < 5) is equivalent to P(X \(\leq\) 4).

Probabilities of the Form P(X \(\geq\) x) or P(X > x)

Probabilities of the form P(X \(\geq\) x) or P(X > x) make use of the complement rule and the ppois command.

For example, suppose \(\mu\) = 10. Let’s compute P(X > 16 ). To do this, we recognize that

P(X > 16) = 1 - P(X \(\leq\) 16)

1 - ppois(16,10)
## [1] 0.02704161

So, P(X > 16) = 0.0270.