The outcome of an experiment need not be a number, for example, the outcome when a coin is tossed can be 'heads' or 'tails'. However, we often want to represent outcomes as numbers. A random variable is a function that associates a unique numerical value with every outcome of an experiment. The value of the random variable will vary from trial to trial as the experiment is repeated.
A random variable has either an associated probability distribution (discrete random variable) or probability density function (continuous random variable).
- A coin is tossed ten times. The random variable X is the number of tails that are noted. X can only take the values 0, 1, ..., 10, so X is a discrete random variable.
- A light bulb is burned until it burns out. The random variable Y is its lifetime in hours. Y can take any positive real value, so Y is a continuous random variable.
The probability distribution of a discrete random variable is a list of probabilities associated with each of its possible values. It is also sometimes called the probability function or the probability mass function.
- More formally, the probability distribution of a discrete random variable X is a function which gives the probability p(xi) that the random variable equals xi, for each value xi: p(xi) = P(X=xi)
It satisfies the following conditions:
Cumulative Distribution Function
All random variables (discrete and continuous) have a cumulative distribution function. It is a function giving the probability that the random variable X is less than or equal to x, for every value x.
- Formally, the cumulative distribution function F(x) is defined to be:
For a discrete random variable, the cumulative distribution function is found by summing up the probabilities as in the example below.
For a continuous random variable, the cumulative distribution function is the integral of its probability density function.
- Discrete case : Suppose a random variable X has the following probability distribution p(xi):
xi 0 1 2 3 4 5 p(xi) 1/32 5/32 10/32 10/32 5/32 1/32
- This is actually a binomial distribution: Bi(5, 0.5) or B(5, 0.5). The cumulative distribution function F(x) is then:
xi 0 1 2 3 4 5 F(xi) 1/32 6/32 16/32 26/32 31/32 32/32
- F(x) does not change at intermediate values. For example:
- F(1.3) = F(1) = 6/32
- F(2.86) = F(2) = 16/32