Bernoulli Trials and Binomial Distribution

Discrete Random Variables and Distributions

Discrete Random Variables:

A discrete variable is a variable which can only take a countable number of values.

Illustration:

If a coin is tossed three times, the number of heads obtained can be 0, 1, 2 or 3. The probabilities of each of these possibilities can be tabulated as shown:

In this example, the number of heads can only take 4 values (0, 1, 2, 3) and so the variable is discrete. The variable is said to be random if the sum of the probabilities is one.

Cumulative Distribution Function:

The cumulative distribution function (c.d.f.) of a discrete random variable X is the function F(t) which tells you the probability that X is less than or equal to t. So if X has p.d.f. P(X = x), we have:

In other words, for each value that X can be which is less than or equal to t, work out the probability that X is that value and add up all such results.

Illustration:

If a die is thrown repeatedly, let’s work out P(X ≤ t) for some values of t.

P(X ≤ 1) is the probability that the number of throws until we get a 6 is less than or equal to 1. So it is either 0 or 1.

P(X = 0) = 0 and P(X = 1) = 1/6. Hence P(X ≤ 1) = 1/6

Similarly, P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0 + 1/6 + 5/36 = 11/36

Probability Density Function:

The probability density function (p.d.f.) of X (or probability mass function) is a function which allocates probabilities. Put simply, it is a function which tells you the probability of certain events occurring. The usual notation that is used is P(X = x) = something. The random variable (r.v.) X is the event that we are considering. So in the above example, X represents the number of heads that we throw. So P(X = 0) means "the probability that no heads are thrown". Here, P(X = 0) = 1/8 (the probability that we throw no heads is 1/8).

In the above example, we could therefore have written:

Quite often, the probability density function will be given to you in terms of x. In the above example, P(X = x) = ³C_x / (2)³

Mean and Variance of Random Variables

Mean:

The mean of a discrete random variable X is a weighted average of the possible values that the random variable can take. Unlike the sample mean of a group of observations, which gives each observation equal weight, the mean of a random variable weights each outcome x_i according to its probability, p_i. The common symbol for the mean (also known as the expected value of X) is μ, formally defined by

μ_x = x₁p₁ + x₂p₂ +...+ x_kp_k

= ∑x_ip_i

The mean of a random variable provides the long-run average of the variable, or the expected average outcome over many observations.

Variance:

The variance of a discrete random variable X measures the spread, or variability, of the distribution, and is defined by

The standard deviation σ is the square root of the variance.