Bernoulli, Binomial and Poisson Random Variables

The first paragraph introduces the concept of a Bernoulli random variable, which represents a simple experiment with only two possible outcomes: success or failure. The classic example used is flipping a coin, where heads is considered a success and tails a failure. The random variable X is defined as the number of successes in a single trial. The probabilities of success (P) and failure (Q, where Q = 1 - P) are the key parameters of the Bernoulli distribution. The expected value (mean) of X is P, and the variance is P*Q, indicating the deviation from the mean. The paragraph also touches on the implications of a fair or biased coin affecting these probabilities.

The second paragraph delves into the binomial distribution, which is an extension of the Bernoulli experiment. It involves repeating the Bernoulli experiment a fixed number of times (n), and X is the total number of successes observed. The binomial distribution has two parameters: n, the number of trials, and P, the probability of success in each trial. The probability of obtaining exactly X successes is calculated using combinations (n choose X), multiplied by the probability of success (P) raised to the power of X and the probability of failure (Q) raised to the power of (n - X). The mean and variance of a binomial random variable are nP and nPQ, respectively. The paragraph also provides an example of calculating the probability of getting a certain number of heads when flipping a coin ten times.

This paragraph explains that a binomial experiment can be viewed as the sum of n Bernoulli random variables, each representing a single trial with its own success or failure outcome. Since the trials are independent, the mean of the binomial variable is simply the sum of the means of the individual Bernoulli variables, which is nP. Similarly, the variance of the binomial variable is the sum of the variances of the Bernoulli variables, resulting in nPQ. This understanding simplifies the calculation of the mean and variance for a binomial random variable.

The fourth paragraph introduces the Poisson distribution, which is used to model the number of events occurring in a fixed time interval. The parameter lambda (λ) represents the average rate of events occurring per time interval. The mean and variance of a Poisson random variable are both equal to lambda. The probability of observing X events in a given time interval is calculated using the formula e^(-lambda) * (lambda^X) / X!, where e is the base of the natural logarithm. An example is provided where someone counts the number of black cars passing by in a 10-minute interval, with an average of six black cars observed, and the probability of observing eight black cars is calculated.

The final paragraph acknowledges the complexity of proving the formulas for the Poisson distribution and suggests that they will be accepted without proof. It emphasizes the practical use of these formulas in various applications, such as counting the number of cars or other events over a fixed time period, and provides a brief overview of the process involved in calculating probabilities using the Poisson distribution.