Chapter 4: Discrete Random Variables
Two Types of Random Variables
Random variable: A variable that assumes numerical values that are determined by the outcomes of an experiment,
where one and only one numerical value is assigned to each experimental outcome.
Discrete random variable:
o A random variable that takes on a finite number or countable infinite number of values
Continuous random variable:
o A random variable that assumes any numerical value in one or more interval on the real number line.
Discrete Probability Distributions
Probability distribution (Of a random variable): A table, graph, or formula that gives the probability associated with
each possible value that a random variable can assume.
o Probability distribution of a discrete random variable x is denoted as p(x)
o P(x) 0 for each value of x.
o ∑ ( )
Mean of a discrete random variable: ∑ ( )
o The value expected to occur in the long run
Variance of a discrete random variable: ∑ ( ) ( )
o Measures the spread of the values of the random variable from their expected value.
Standard deviation of a discrete random variable:
o Positive squared root of the variance.
o Chebyshev’s theorem application:
o For every integer k 1, p(x) falls in the interval [ ] 1 – 1/k
o The experiment consists of (n) identical trials.
o Each trial results in a success or a failure.
o The probability of a success on any trial is p, and the remains constant from trial to trail. This implies that the
probability of failure, q, on any trial is 1- p, and remains constant from trial to trail.
o The trials are independent (each result has no relation to each other)
Binomial random variable (x): The total number of successes in n trials of a binomial experiment
Binomial formula: ( ) ( ) ⁄ ( )
o The probability of getting x successes in n trials
o = The probability of getting x successes in a particular combination
o The number of ways to arrange x successes in n trials
⁄ ( )
Binomial table: Tables in which we can look up binomial probabilities