7.1 Discrete and Continuous Random Variables A random variable is:

• We usually denote random variables by _____________________ such as X or Y • When a random variable X describes a random phenomenon, the ___________________ S just lists the _________________ of the random variable. • Example:

Let X = count of heads in 4 tosses X P(x)

• What is the probability distribution of the discrete random variable X that counts the number of heads in four tosses of a coin? • Probability of tossing at least 2 heads? • Probability of at least one head?

Discrete random variable A discrete random variable X has: The probability distribution of a discrete random variable X lists:

The probabilities must satisfy 2 requirements: 1) 2)

Example • The instructor of a large class gives 15% each of A’s and D’s, 30% each of B’s and C’s, and 10% F’s. Choose a student at random from this class. The student’s grade on a 4-pt scale (A = 4) is a random variable X. Find the probability that the student got a B or better.

You! Construct the probability distribution for the number of boys in a three-child family. Find the following probabilities: 1) P(2 or more boys) •

2) P(No boys) 3) P(1 or less boys)

In an article in the journal Developmental Psychology (March 1986), a probability distribution for the age X (in years) when male college students began to shave regularly is shown:

Here is the probability distribution for X in table form:

X

11

12 13

14

15

16

17

18

19

20+

P(x)

0.013

0

0.067

0.213

0.267

0.240

0.093

0.067

0.013

0.027

1) Is this a valid probability distribution? What is the random variable of interest? Is X discrete? 2) What is the most common age at which a randomly selected male college student begins shaving? 3) What is the probability that a randomly selected male college student begins shaving at 16? What is the probability that a randomly selected male college student begins shaving before 15?

Continuous Random Variables A continuous random variable X:

Example: S = {all numbers x between 0 and 1 inclusive}

.

• The probability distribution of X assigns probabilities as ___________________________  • Any density curve has area exactly ___ underneath it (probability = ___)

Example • A random number generator will • Find the probability that the generator produces a spread its output uniformly number X between 3 and 7 across the entire interval from 0 to 9 as we allow it to generate a long sequence of numbers. The • Find the probability that results of many trials are the generator produces a represented by the density curve number X less than or of a ____________________. equal to 5 or greater than 8

Special Note: • All continuous probability distributions assign probability ____ to every _________________. • The probability of x __.8 is the same as x __ .8 Example: Find P(.79 < x < .81) = Find P(.799 < x < .801) = Find P(.7999 < x < .8001) = Find P(x=.8) =

Normal Distributions as Probability Distributions • Because any density curve describes an assignment of probabilities, ________________________________________. • If X has the N(µ ,σ) distribution, then z=

x−µ

σ is a ______________________________ having the distribution N(0,1).

Example • An opinion poll asks an SRS of 1500 adults what they consider to be the most serious problem affecting schools. Suppose that if we could ask all adults this ?, 30% would say “drugs.” • Assume your sample proportion follows a normal distribution: N(.3, .0118).

• Given: Mean = .3, and Standard dev. = .0118 • Find the probability that the poll result differs from the truth about the population by more than 2 percentage points.

1) The probabilities that a randomly selected customer purchases 1, 2, 3, 4, or 5 items at a convenience store are .32, .12, .23, .18, and .15, respectively. a) Identify the random variable of interest. X = ____. Then construct a probability distribution (table), and draw a probability distribution histogram. b) Find P(X>3.5) c) Find P(1.0