Random Variable: A random Variable (r.v.) is a numerical quantity the value of which is determined by an experiment – human or natural. In other words, its value is determined by chance.

Experiment : Man-made (tossing a coin) or natural (Albany temperature tomorrow).

Sample Space: All possible outcomes of the experiment. The random value must assume one or more of these outcomes.

Illustrations:

Tossing of a coin – experiment.

{H ,T} – sample space.

We can define our r.v. as "occurrence of H".

Another Illustration:

r.v.–"the sum of numbers showing on two dice".

Experiment: Throwing a pair of dice.

Sample Space : See figure 4.1 (page 127).

Three points about r.v.:
r.v. must assume numerical values. Qualitative outcomes can be assigned numerical A values like 0 and 1.

The value of a r.v. must be defined for all possible outcomes of the experiment in question, i.e., for elements of the sample space.

Value of the r.v. is unknown before the experiment in question is carried out. After the experiment is carried out, the value of the r.v. is always known.

Discrete and Continuous Random Variables:

A discrete r.v. takes only a finite or countable number of distinct values. Outcomes from tossing a coin.

A Continuous r.v. takes any numerical value on a continuous scale. Example: Income of a family.

Probability Distribution: The probability distribution of a r.v. X provides the probability of each possible value of a random variable.

If P(x) is the probability that x is the value of the r.v., we can be sure that S P(x) = 1, where the summation is over all values that X takes on. This is because these values are mutually exclusive, and one of them must occur.

Note the subtle distinction between X and x. X is the random variable and x is a specific value assumed by X.

Illustration of a Probability Distribution:

Sum of the numbers showing on two dice.

See Table 4.1 (page 131).

See also Figure 4.2 (page 131) for a line chart.

Relative Frequency Distibution:

A relative frequency distribution shows the proportion (not the number) of cases falling within each class interval.

Consider the number coming up in a single die.

P(x) = 1/6 for x = 1, 2, 3, 4, 5,6.

See Table 4.2 (page 133).

Probability distributions are often used to represent or approximate population relative frequency distribution.

Expected Value

The expected value of a discrete r.v. X, denoted by E(X) is the weighted mean of the possible values that the r.v. can assume, where the weight attached to each value is the probability that a r.v. will assume this value. In other words,

E(X) = S x_iP(x_i)

Where the r.v. X can assume m possile values x₁,x₂, …, x_m and the probability of its value equaling x_i is P(x_i ).

It I like a weighted mean of a frequency distribution. So it is also called mean.

Example: Rolling a die. Suppose you get $1 for face with one dot, $2 for a face with two dots, etc.

If you throw once what is the expectedpay off?

E(X) = S x_i P(x_i ).

=$1x1/6 + $2x1/6 + ..+$6x1/6

= $ 3.5

Expected value has important use in decision making.