Determine the indicated probability for a Poisson random variable with the given values of $\lambda \text { and } t \text {. }$ $\lambda = 0.9 , t = 3 , P ( \text { Less than } 3 )$
A) 0.6469
B) 0.7141
C) 0.4936
D) 0.5064

Question

Quizplus · Accepted Answer

To calculate this probability we can use the cumulative distribution function (CDF) of a Poisson random variable with parameter $\lambda$, which is given by:
$$F(k;\lambda)=\sum_{i=0}^ke^{-\lambda}\frac{\lambda^i}{i!}$$
This represents the probability of observing at most k events in a unit of time or space, when events occur at a rate of $\lambda$ per unit.

In this case, we have $\lambda=0.9$ and $t=3$, which means that the expected number of events in 3 units of time or space is $\mu=\lambda t=2.7$.

We want to calculate $P(X<3)$, which is equivalent to $P(X\leq 2)$, since X is a discrete random variable. Using the Poisson CDF, we have:
$$P(X\leq 2)=F(2;\mu)=\sum_{i=0}^2e^{-\mu}\frac{\mu^i}{i!}=e^{-\mu}\left(1+\mu+\frac{\mu^2}{2}\right)\approx 0.4936$$
Therefore, the best choice is C, with a probability of approximately 0.4936.

Determine the Indicated Probability for a Poisson Random Variable with the Given