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Warm Up: 2003 AP FRQ #2. 7.1 Discrete and Continuous Random Variables. We usually denote random variables by capital letters such as X or Y When a random variable X describes a random phenomenon, the sample space S just lists the possible values of the random variable.

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slide2

7.1 Discrete and Continuous Random Variables

  • We usually denote random variables by capital letters such as X or Y
  • When a random variable X describes a random phenomenon, the sample space S just lists the possible values of the random variable.
  • Example: The count of heads in four tosses of a coin
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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?
example
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.
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You!
  • Construct the probability distribution for the number of boys in a three-child family. Find the following probabilities:
  • P(2 or more boys)
  • P(No boys)
  • P(1 or less boys)
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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:

  • Is this a valid probability distribution? What is the random variable of interest? Is X discrete?
  • What is the most common age at which a randomly selected male college student begins shaving?
  • 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
Example:

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

The probability distribution of X assigns probabilities as area under a density curve 

Any density curve has area exactly 1 underneath it (probability = 1)

.

Continuous Random Variables
example1
A random number generator will spread its output uniformly across the entire interval from 0 to 9 as we allow it to generate a long sequence of numbers. The results of many trials are represented by the density curve of a uniform distribution.

Find the probability that the generator produces a number X between 3 and 7

Find the probability that the generator produces a number X less than or equal to 5 or greater than 8

Example
special note
Special Note:
  • All continuous probability distributions assign probability 0 to every individual outcome.

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
Normal Distributions as Probability Distributions
  • Because any density curve describes an assignment of probabilities, normal distributions are probability distributions.
  • If X has the N( ) distribution, then

is a standard normal random variable having the distribution N(0,1).

example2
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 q, 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.

Example
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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 <X<3.0)

d) Find P(X<5)

2) A certain probability density function is made up of two straight-line segments. The first segment begins at the origin and goes to the point (1,1). The second segment goes from (1,1) to the point (x, 1).

a) Sketch the distribution function, and determine what x has to be in order to be a legitimate density curve.

b) Find P(0<X<.5)

c) Find P(X=1)

d) Find P(0<X<1.25)

e) Circle the correct option: X is an example of a (discrete) (continuous) random variable.