SWBAT: Use a probability distribution to calculate the mean
and standard deviation of a discrete random variable.
Lesson 5-6
Do Now:
1.
2.
SWBAT: Use a probability distribution to calculate the mean
and standard deviation of a discrete random variable.
Lesson 5-6
Random Variable: Takes numerical values that describe the outcomes of some event
Probability Distribution: The probability distribution of a random variable gives its possible
values and their probabilities.
Discrete Random Variable: a variable that takes on fixed values as possible outcomes in an
event
Probability Distribution for a discrete random variable
Every probability Pi is a number between 0 and 1.
The sum of the probabilities is 1: P1 + P2 + P3 +…= 1.
Example 1: (reading a probability distribution)
North C arolina State University posts the grade distributions for its courses online. Students in
Statistics 101 in a recent semester received 26% As, 42% Bs, 20% C s, 10% Ds, and 2%
Fs. C hoose a Statistics 101 student at random. The student’s grade on a four-point scale
(with A = 4) is a discrete random variable X with this probability distribution:
(a) Describe, in words, what the meaning of P(X ≥ 3) is. What is this probability?
(b) Write the event “the student got a grade worse than C ” in terms of values of the random
variable X. What is the probability of this event?
SWBAT: Use a probability distribution to calculate the mean
and standard deviation of a discrete random variable.
Lesson 5-6
Mean (Expected Value) of a Discrete Random Variable
To find the mean (expected value) of X, multiply each possible value by its probability, then add
all the products:
Standard deviation of a random variable
The square root of the variance of a random variable
variability of the distribution about the mean.
. The standard deviation measures the
SWBAT: Use a probability distribution to calculate the mean
and standard deviation of a discrete random variable.
Lesson 5-6
Example 2: (calculating the mean [expected value] of a discrete random variable)
In 1952, Dr. Virginia Apgar suggested five criteria for measuring a baby’s health at birth: skin
color, heart rate, muscle tone, breathing, and response when stimulated. She developed a 0-1-2
scale to rate a newborn on each of the five criteria. A baby’s Apgar score is the sum of the
ratings on each of the five scales, which gives a whole-number from 0 to 10. Apgar scores are
still used today to evaluate the health of newborns.
What Apgar scores are typical? To find out, researchers recorded the Apgar scores of over 2
million newborn babies in a single year. Imagine selecting one of these newborns at
random. (That’s our chance process.) Define the random variable X = Apgar score of a randomly
selected baby one minute after birth. The table below gives the probability distribution for X.
(a) Doctors decided that Apgar scores of 7 or higher indicate a healthy baby. What’s the
probability that a randomly selected baby is healthy?
(b) C ompute the mean of the random variable X and interpret this value in context.
(c) C ompute and interpret the standard deviation of the random variable X.
SWBAT: Use a probability distribution to calculate the mean
and standard deviation of a discrete random variable.
Lesson 5-6
You Try!!
A large auto dealership keeps track of sales made during each hour of the day. Let X = the
number of cars sold during the first hour of business on a randomly selected Friday. Based on
previous records, the probability distribution of X is as follows:
(a) C ompute and interpret the mean of X.
(b) C ompute and interpret the standard deviation of X.
SWBAT: Use a probability distribution to calculate the mean
and standard deviation of a discrete random variable.
Lesson 5-6
LESSON PRACTICE
1. Spell-checking software catches “nonword errors,” which result in a string of letters that is not
a word, as when “the” is typed as “teh.” When undergraduates are asked to write a 250-word
essay (without spell-checking), the number X of nonword errors has the following distribution:
(a) Write the event “at least one nonword error” in terms of X. What is the probability of this
event?
(b) Describe the event X ≤ 2 in words. What is its probability? What is the probability that
X < 2?
2. C hoose a person aged 19 to 25 years at random and ask, “In the past seven days, how many
times did you go to an exercise or fitness center or work out?” C all the response Y for
short. Based on a large sample survey, here is a probability model for the answer you will get:
(a) Show that this is a legitimate probability distribution.
(b) Describe the event Y < 7 in words. What is P(Y < 7)?
(c) Express the event “worked out at least once” in terms of Y. What is the probability of this
event?
SWBAT: Use a probability distribution to calculate the mean
and standard deviation of a discrete random variable.
Lesson 5-6
3. C hoose a person aged 19 to 25 years at random and ask, “In the past seven days, how many
times did you go to an exercise or fitness center or work out?” C all the response Y for
short. Based on a large sample survey, here is a probability model for the answer you will get:
(a) Describe the event Y < 7 in words. What is P(Y < 7)?
(b) Express the event “worked out at least once” in terms of Y. What is the probability of this
event?
(c) C alculate the mean of the random variable X and interpret this result in context.
(d) C alculate and interpret the standard deviation of the random variable X.
SWBAT: Use a probability distribution to calculate the mean
and standard deviation of a discrete random variable.
Lesson 5-6
4. In an experiment on the behavior of young children, each subject is placed in an area with
five toys. Past experiments have shown that the probability distribution of the number X of toys
played with by a randomly selected subject is as follows:
(a) Write the event “plays with at most two toys” in terms of X. What is the probability of this
event?
(b) Describe the event X > 3 in words. What is its probability? What is the probability that
X ≥ 3?
(c) C alculate the mean of the random variable X and interpret this result in context.
(d) C alculate and interpret the standard deviation of the random variable X. Show your work.