The random variable X follows a Poisson process with the...

The random variable $X$ follows a Poisson process with the given value of $\lambda$ and $t.$
Assuming $\lambda=0.02$ and $t=50,$ compute
(a) $P(2)$
(b) $P(X<2)$
(c) $P(X \geq 2)$
(d) $P(1 \leq X \leq 3)$
(e) $\mu_{X}$ and $\sigma_{X}$

Key Concepts

Statistical Moments: Mean and Variance

For a Poisson distribution, both the mean and the variance are equal to ?t. This property is unique to the Poisson process and indicates that the spread of the distribution (variance) is directly tied to the average rate of occurrence, simplifying many analytical processes involving the distribution.

Poisson Distribution

The Poisson distribution is used to model the probability of a given number of events occurring in a fixed interval of time or space, provided these events occur with a constant mean rate and independently of the time since the last event. It is commonly applied in scenarios where events happen randomly and independently.

Probability Mass Function (PMF)

The probability mass function for a Poisson distribution provides the probability of observing exactly k events. It is defined by the formula P(X = k) = (?t)^k * e^(-?t) / k!, where ? is the rate of occurrence and t is the time period. This function is fundamental for calculating specific probabilities such as P(2).

Cumulative Distribution Function (CDF)

The cumulative distribution function for a Poisson distribution sums the probabilities of the random variable being less than or equal to a specific value. It is particularly useful for determining probabilities over a range of events, such as P(X < 2), which involves summing the probabilities for all outcomes less than 2.

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