Probability

To find the probability that a random variable X, which follows a Poisson distribution with a mean (λ) of 5 , is less than 3 , we calculate P(X<3). This is equivalent to finding the sum of probabilities P(X=0),P(X=1), and P(X=2) :
P(X<3)‌=P(X=0)+P(X=1)+P(X=2)
‌=‌

λ0e−λ

0!

+‌

λ1e−λ

1!

+‌

λ2e−λ

2!

Substituting λ=5, we have:
P(X<3)‌=e−5(1+5+‌

25

2

)
‌=e−5(‌

2

2

+‌

10

2

+‌

25

2

)
‌=e−5×‌

37

2

‌=‌

37

2

e−5