Chapter 5: The Binomial Probability Distribution and Related Topics Section 1: Introduction to Random Variables and Probability Distributions

 The quantitative variable x is a random variable because the value that x takes on in a given experiment is a chance or random outcome.  Continuous Random Variable – has an infinite number of possible values that is not countable; usually measures the amount of something  Discrete Random Variable - has either a finite number of possible values or a countable number of possible values; usually counts something

Examples 1. The number of A’s earned in a math class of 15 students (Discrete) (Discrete) 2. The number of cars that travel through McDonald’s drive thru in the next hour (Discrete) 3. The speed of the next car that passes a state trooper (Continuous) (Continuous)

 probability distribution – an assignment of probabilities to specific values of the random variable Probability Distribution xP(x) xP(x) 0 ¼ 1 ½ 2 ¼

Properties of Discrete Probability Distributions  0 < P(x) < 1  The sum of the values of P(x) for each distinct value of x is 1. ∑P(x) = 1 ∑P(x) = 1 xP(x) xP(x) Which of the following is a probability distribution?xP(x) A.B.C.

Determine the required value of the missing probability in order to make the distribution a discrete probability distribution. A. B. xP(x) xP(x)

 Given a discrete random variable x with probability distribution P(x), the mean of the random variable x is defined as = ∑x ∙ P(x) = ∑x ∙ P(x)

= ∑ (x - )2∙P(x) = ∑ (x - )2∙P(x)  Given a discrete random variable x with probability distribution P(x), the variance of the random variable x is defined as *The standard deviation is the square root of the variance.

Example For the following probability distribution, find the mean, variance, and standard deviation. xP(x) 14/10 24/10 31/10