4.2 Probability Distributions

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1 4.2 Probability Distributions Definition. A random variable is a variable whose value is a numerical outcome of a random phenomenon. The probability distribution of a random variable tells us what the possible values of the variable are and how probabilities are assigned to those values. Discrete Random Variables Definition. A discrete random variable X has a finite number of possible values. The probability distribution of X lists the values and their probabilities: Example 4.9. A household is a group of people living together, regardless of their relationship to each other. Many sample surveys such as the Current Population Survey select a random sample of households. Choose a household at random, and let the random variable X be the number of people living there. Here is the distribution of X. Household size Probability The probability that a randomly chosen household has more than two 1

2 members is P (X >2) = P (X =3)+P (X =4)+P (X =5)+P (X =6)+P (X =7) = =.428 Equally Likely Outcomes Definition. If a random phenomenon has k possible outcomes, all equally likely, then each individual outcome has probability 1/k. The probability of any event A is P (A) = = count of outcomes in A count of all possible outcomes count of outcomes in A. k Example Roll two dice and record the pips (dots) on each of the two up-faces. Figure 4.8 (see TM-65) shows the 36 possible outcomes. If the dice are carefully made, all 36 outcomes are equally likely. So each has probability 1/36. Gamblers are often interested in the sum of the pips on the up faces. What is the probability of rolling a 5? The event roll a 5 contains the four outcomes: (1,4), (2,3), (3,2), (4,1). The probability is therefore 4/36 = 1/9 = What about the probability of rolling a 7? In Figure 4.8 (TM-65) you will find six outcomes for which the sum of the pips is 7. The probability is 6/36 = 1/6 =

3 The Mean and Standard Deviation of a Discrete Random Variable Definition. Suppose that X is a discrete random variable whose distribution is: Find the mean of X by multiplying each possible value by its probability and adding over all the values: µ = x 1 p 1 + x 2 p x k p k = n x i p i. i=1 Note. The mean of a random variable X is a single fixed number µ. It gives the average value of X in several senses: The mean µ is the average of the possible values of X, each weighted by how likely it is to occur. That s what the definition of µ says. The mean µ is the point at which the probability histogram of the distribution of X would balance if made of solid material. See Figure 4.9 (and TM-66). Recall that the mean µ of a density curve has this same property. If we actually repeat the random phenomenon many times, record the value of X each time, and average these observed values, this average will get closer and closer to µ as we make more and more repititions. This fact is called the law of large numbers. 3

4 Definition. Suppose that X is a discrete random variable whose distribution is: and that µ is the mean of X. Thevariance of X is σ 2 =(x 1 µ) 2 p 1 +(x 2 µ) 2 p 2 + +(x k µ) 2 p k = n (x i µ) 2 p i. The standard deviation σ is the square root of the variance. i=1 Continuous Random Variables Definition. A continuous random variable X takes all values in an interval of numbers. The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that make up the event. Note. The distribution of a continuous random variable assigns probabilities as areas under a density curve. See Figure 4.10 (and TM-67). Definition (for those with some calculus background). Suppose that X is a continuous random variable with probability distribution P (X). The mean of X is µ = xp (x) dx and the variance of X is σ 2 = (x µ) 2 P (x) dx, 4

5 where the integrals are taken over all possible values of X. Thestandard deviation σ is the square root of the variance. 5

AP Statistics Chapter 6 - Random Variables

AP Statistics Chapter 6 - Random 6.1 Discrete and Continuous Random Objective: Recognize and define discrete random variables, and construct a probability distribution table and a probability histogram