Binomial Distribution and Poisson Distribution

Binomial Distribution

The Binomial Distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials(a random experiment with exactly two possible outcomes), each with the same probability of success. This distribution is used in situations where there are only two possible outcomes, often referred to as "success" and "failure".

Formula:

Mathematically, if X is a binomial random variable representing the number of successes in n trials, with probability of success p, the probability mass function (PMF) is given by:

$P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$

where:

$\binom{n}{k}$ is the binomial coefficient, calculated as $\frac{n!}{k!(n-k)!}$

n is the number of trials

k is the number of successes

p is the probability of success in a single trial

Example:

Suppose you flip a fair coin 10 times. The probability of getting exactly 6 heads can be calculated using the binomial distribution with n = 10, p = 0.5, and k = 6.

Using the formula:

$P(X = 6) = \binom{10}{6} (0.5)^6 (0.5)^4$

Calculating the values:

$\binom{10}{6} = \frac{10!}{6!4!} = 210$

So,

$P(X = 6) = 210 \times (0.5)^6 \times (0.5)^4 = 210 \times (0.015625) \times (0.0625) = 0.205$

Therefore, the probability of getting exactly 6 heads in 10 flips of a fair coin is 0.205, or 20.5%.

Characteristics:

The binomial distribution has two key parameters:

n: the number of trials

p: the probability of success in each trial

The mean (expected value) of a binomial distribution is $\mu = np$

The variance of a binomial distribution is $\sigma^2 = np(1 - p)$

Poisson Distribution

The Poisson Distribution is a discrete probability distribution that expresses the probability of a given number of events occurring within a fixed interval of time or space, provided these events happen with a known constant mean rate and independently of the time since the last event. It is used for modeling the number of times an event occurs in a given interval of time or space.

Formula:

Mathematically, if X is a Poisson random variable representing the number of events, with mean rate $\lambda$, the probability mass function (PMF) is given by:

$P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$

where:

$\lambda$ is the average rate of occurrence

$e$ is the base of the natural logarithm (approximately equal to 2.71828)

k is the number of occurrences

Example:

Suppose a call center receives an average of 5 calls per hour. What is the probability that exactly 3 calls will be received in a given hour?

Using the formula with $\lambda = 5$ and k = 3:

$P(X = 3) = \frac{5^3 e^{-5}}{3!}$

Calculating the values:

$e^{-5} \approx 0.0067$

So,

$P(X = 3) = \frac{125 \times 0.0067}{6} \approx 0.1404$

Therefore, the probability of receiving exactly 3 calls in an hour is approximately 0.1404, or 14.04%.

Characteristics:

The Poisson distribution has one key parameter:

$\lambda$: the average rate of occurrence

The mean (expected value) of a Poisson distribution is $\mu = \lambda$

The variance of a Poisson distribution is also $\sigma^2 = \lambda$