Random Variables and Probability Distributions

Definition A random variable is a real-valued function whose domain is the sample space for some experiment.

Example 1 When a coin is tossed 3 times, let X be the number of 'Heads' obtained. Then X is a function which maps the eight sample- points to the real numbers {0,1,2,3}.

Example 2 When a pair of dice is thrown, let X be the sum of the numbers of spots on the two faces. Then X is a function which maps the set of 36 possible outcomes to the set of real numbers {2,3,4,5,6,7,8,9,10,11,12}.

Example 3 When a point is chosen at random from inside a circle of radius R, let X be its distance from the circumference. Then X is a function which maps the infinite set of number-pairs (x,y) satisfying x 2 + y 2 < r 2 to the interval of real numbers satisfying 0 < r ≤ R.

Example 4 Suppose values are chosen at random, one after another, from the unit-interval [0,1] until their sum has exceeded 1.0, and let X count the number of values that get added during this experiment, so that X is a random variable. What will be the possible values for X ? And what set of 'sample-points' describes this experiment?

Definition Because a random variable X is a function on the sample space S for some experiment, say S = { x 1, x 2, x 3,..., x n }, where each sample-point x k has a probability p k associated with it, there will be a probability distribution function f induced by the random variable X, defined by f( x k ) = p k = P( X = x k ).

Properties of f Any distribution function f associated with a random variable X will have two properties: (1)f( x k ) ≥ 0, (for k=1,2,...,n) (2)f( x 1 ) + f( x 2 ) f( x n ) = 1.

Probability graph of f X f X = Number of 'Heads' when a coin is tossed 3 times 3/8 2/8 1/8

Amother probability graph X f X = Number of spots obtained when two dice are thrown

Mean of a random variable If X is a random variable defined on a sample space S = { x 1, x 2,..., x n } and f is its associated distribution function, then the average value for X (called the mean of X) is given by the formula μ X = x 1 f(x 1 ) + x 2 f(x 2 ) x n f(x n )

Example 1 For the experiment of tossing a coin 3 times, the probability distribution is given by this table: Number of 'Heads' Probability f : 1/83/83/81/8 So the mean number of heads is μ X = 0(1/8) + 1(3/8) + 2(3/8) + 3(1/8) = 12/8 = 1.5

Example 2 For the experiment of throwing a pair of dice, the probability distribution is: Number of spots: Probability f :1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 So the mean number of spots will be given by the following weighted average: (2)(1) + (3)(2) + (4)(3) + (5)(4) + (6)(5) + (7)(6) + (8)(5) + (9)(4) + (10)(3) + (11)(2) + (12)(1) 36 = ( )/36 = 252/36 = 7