Examples of discrete random variables include the number of children in a family, the Friday night attendance at a cinema, the number of patients in a doctor’s surgery, the number of defective light bulbs in a box of ten.
It is not possible to observe values that are real numbers, such as 2.091.
(Remark: it is possible for the average of a discrete random variable to be a real number.)
For a six sided dice, the only possible observed values are 1, 2, 3, 4, 5 and 6. It is not possible to observe values such as 5.732.
For a continuous random variable all possible fractional values of the variable cannot be listed, and therefore the probabilities that are determined by a mathematical function are portrayed graphically by a probability density function, or probability curve.
The expected value of a random variable X is symbolised by E(X) or \(mu\).
If X is a discrete random variable with possible values \(\{ x1, x2, x3,\ldots , xn\}\), and$ p(x_i)$ denotes P(X = xi), then the expected value of X is defined by:
\[E(X) = \sum x_i \times P(x_i) \]
where the elements are summed over all values of the random variable X.
A random variable is a numerical description of the outcome of an experiment.
Random variables can be classified as discrete or continuous, depending on the numerical values they may take.
A ranom variable that may assume any numerical value in an interval or collection of intervals is called a continuous random variable.
Suppose a fair coin is tossed six times. The number of heads which can occur with their respective probabilities are as follows:
xi0123456 p(xi)1/646/6415/6420/6415/646/641/64
a)Compute the expected value (i.e. expected number of heads). b)Compute the variance of the number of heads.