Discrete random variable A random variable is a variable which takes numerical values and whose value depends on the outcome of an experiment. It is discrete if it can only take certain values. For a discrete random variable X with P(X = x) then probabilities always sum to 1. P(X = x) = 1 (Remember means 'sum of'). Cumulative distribution function 'Cumulative' gives us a kind of running total, so a cumulative distribution function gives us a running total of probabilities within our probability table. The cumulative distribution function, F(x) of X is defined as: Expectation The expectation is the expected value of X, written as E(X) or sometimes as . The expectation is what you would expect to get if you were to carry out the experiment a large number of times and calculate the 'mean'. To calculate the expectation we can use: E(X) = x P(X = x) Expectation of any function of x If X is a discrete random variable and f(x) is any function of x, then the expected value of f(x) is given by: E[f(x)] = f(x)P(X = x) There are a few general results we should remember to help with our calculations of expectations: E(a) = a E(aX + b) = aE(X) + b Variance The variance is a measure of how spread out the values of X would be if the experiment leading to X were repeated a number of times. The variance of X, written as Var(X) is given by: Var(X) = E(X2) - (E(X))2, If we write E(X) = then, Var(X) = E(X2) - 2 Or Var(X) = E(X - )2, this tells us that Var(X) 0 There are a few general results we should remember to help with our calculations of variances: Var(aX) = a2Var(X) Var(aX + b) = a2Var(X) The Standard Deviation The square root of the Variance is called the Standard Deviation of X. Standard deviation is given the symbol . = or Var(X) = 2 The Binomial Distribution Suppose that an experiment consists of n identical and independent trials, where for each trial there are 2 outcomes. 'Success' which is given probability p 'Failure' which is given probability q where q = 1 - p Then if X = the number of successes, we say that X has a binomial distribution. We write: /**/ Sometimes written as: X ~ bin(n, p) If our random variable follows a binomial distribution then the associated probabilities are calculated using the following formula: /**/ gives us the number of ways of choosing r objects from n. It is calculated by: You may also have a button on your calculator that will do all that for you. Expectation and Variance of binomial distribution If: X ~ bin(n, p) Then remember: E(X) = n x p Var(X) = n x p xq n = number of trials p = probability of a success q = probability of a failure = 1 - p Geometric distribution If we let X be the random variable of the number of trials up to and including the first success then X has a Geometric distribution. The probabilities are worked out like this: (Remember, p = probability of success, and q = probability of failure) P(X = 1) = p No failure, success on first attempt (hooray!) P(X = 2) = q x p 1 failure then success P(X = 3) = q2 x p 2 failures then success P(X = 4) = q3 x p 3 failures then success P(X = r) = qr-1 x p r - 1 failures followed by success If X follows a Geometric distribution we write: X ~ Geo(p) This reads as 'X has a geometric distribution with probability of success, p'. A special result to remember: P(X > r) = qr Expectation and variance of geometric distribution If: X ~ Geo(p) Then: E(X) = and Var(X) =