Let X denote a random variable that has a Poisson...

Let $X$ denote a random variable that has a Poisson distribution with mean $\mu=3$ Find the following probabilities, both manually and with a GDC:
a) $P(x=5)$
b) $P(x<5)$
c) $P(x \geqslant 5)$
d) $P(x \geqslant 5 | x \geq 3)$

Key Concepts

Conditional Probability

Conditional probability measures the probability of an event occurring given that another event has already occurred. It is calculated using the formula P(A|B) = P(A ? B) / P(B), where A and B are events. This concept is essential in problems where the outcome is restricted by a given condition, allowing for the adjustment of probability calculations within a subset of the sample space.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, assuming these events happen with a known constant mean rate and independently of the time since the last event. It is widely used to model count-based data in various fields such as queueing theory, telecommunications, and reliability engineering.

Probability Mass Function (PMF)

The probability mass function for a discrete random variable provides the probability that the variable takes a specific value. For the Poisson distribution, the PMF is defined as P(X = k) = (e^(??) * ?^k) / k!, where ? is the average rate (mean) of occurrence and k is a non-negative integer representing the number of occurrences.

Cumulative Distribution Function (CDF)

The cumulative distribution function is a function that gives the probability that a random variable is less than or equal to a certain value. In the context of the Poisson distribution, it is found by summing the probabilities of obtaining 0, 1, 2, ..., up to a given value, helping to compute probabilities for intervals or ranges of outcomes.

Complement Rule

The complement rule is a principle in probability that states the probability of an event not occurring is 1 minus the probability of the event occurring. This rule simplifies the calculation of the probability of compound events such as 'at least' scenarios, for example, P(X ? k) can be computed as 1 ? P(X < k).

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