cumulative distribution functionor just "distribution function" defines the probability that a random variable takes a value equal to or less than a given number: P(X ≤ x) or F(X). Using the probability function defined earlier: X:(1, 2, 3, 4), p(x) = x / 10, F(1) = 0.1, F(2) = 0.3, F(3) = 0.6, F(4) = 1. In other words, F(3) represents the cumulative probability that outcomes 1, 2, and 3 occur.
g: Define a discrete uniform random variable and calculate probabilities, given a discrete uniform probability distribution.
The discrete uniform random variableis one where the probabilities are equal for all possible outcomes. One example is X:(1,2, 3, 4, 5), p(x) = 0.2. In this case, the probabilities are equal for each possible outcome (20%). The probability of any one outcome is 0.2 and the probability of any "n" outcomes is n * 0.2. For example, p(2 ≤ X ≤ 4) = p(2) + p(3) + p(4) = 0.6, and F(2) = p(1) + p(2) = 0.4.
h: Define a binomial random variable and calculate probabilities, given a binomial probability distribution.
The binomial random variableis the number of "success" in a given number of "trial" where the outcome can either be "success" or "failure." The probability of success is constant for each trial, and the trials are independent. A trial is like a mini-experiment and the final outcome is the number of successes in the series of n-trials. Under these conditions, the probability of "x" success in "n" trials is calculated using the following formula:
p(x) = P(X = x) = [number of ways to choose x from n]px(1 - p)n - x