Test Index

Probability

Section: Mathematics

Question : 91 of 92

Marks: +1, -0

X

X < 3

Solution:

To find the probability that a random variable

X

, which follows a Poisson distribution with a mean

(\lambda)

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)

\;=\;\frac{\lambda^0 e^{-\lambda}}{0!}+\;\frac{\lambda^1 e^{-\lambda}}{1!}+\;\frac{\lambda^2 e^{-\lambda}}{2!}

Substituting

\lambda=5

, we have:

P(X < 3)\;=e^{-5} \left(1+5+\;\frac{25}{2}\right)

\;=e^{-5} \left(\;\frac{2}{2}+\;\frac{10}{2}+\;\frac{25}{2}\right)

\;=e^{-5} \times \;\frac{37}{2}

\;=\;\frac{37}{2} e^{-5}

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