Download Introductory Statistics on the TI-83

Chapter
5
Discrete Random
Variables and Their
Probability Distributions
Mean and Standard Deviation of a Discrete Random Variable
Computing the mean and standard deviation of a discrete random variable is slightly different
than computing the mean and standard deviation of a set of data values. Each data value in the
data set weighs equally in the computation. However, in a discrete random variable, the possible
data values are given along with the likelihood of each value occurring on any given single trial.
As was the case for a set of data values, the TI-84 calculator can be used to calculate the mean
and standard deviation of a discrete random variable by either manually using the formulas or by
using a built-in function. We will begin with manually using the formulas.
Example: Number of Breakdowns Per Week
Example 5-3 gives the probability distribution of the number of
breakdowns per week for a machine based on past data. Enter the
number of breakdowns into a list named X, and the probability into a
list named PROBX.
2
Graphing Calculator Manual
Mean of a Discrete Random Variable
The formula to calculate the mean of a discrete
probability distribution is
.
Move the cursor to highlight the name of the empty
List next to List PROBX.
Type: List X * List PROBX
Press ENTER.
The formula now says to sum of this list of values.
Go to the homescreen (2nd > Mode).
Select 2nd > STAT > MATH > 5: sum(
Type: L1) and press ENTER.
µ = 1.8 breakdowns per week.
Standard Deviation of a Discrete Random Variable
The formula to calculate the standard deviation of
a discrete probability distribution is
Move the cursor to
highlight the name of
the empty List next to
List PROBX.
Type: List X ^ 2 * List PROBX
Press ENTER.
Chapter 5: Discrete Random Variables and Their Probability Distributions
3
The formula now says to take the square root of the difference of the sum of this
list of values and the square of the mean.
Go to the homescreen (2nd > Mode).
Press √ Key.
Select 2nd > STAT > MATH > 5: sum(
Type: L1) – 1.8 ^ 2) and press ENTER .
σ = 1.03 breakdowns per week.
Using TI-84 Plus Built-In Functions For Discrete Probability Distributions
The TI-84 Plus built-in function 1-Var Stats will also calculate
the numerical descriptive statistics for a Discrete Probability
Distribution.
We will use the same
probability
distribution
as
above, which we stored in Lists
X and PROBX.
Select STAT > CALC > 1-Var Stats.
Press ENTER.
Select 2nd > STAT >X , 2nd > STAT > PROBX
Press ENTER.
The screen will display the descriptive statistics, which includes
the population mean and standard deviation.
Generating Dependent Probabilities
Factorials
A common function needed to compute dependent probabilities is the factorial function. The
notation for the factorial of n is n! The “!” function is found on the MATH page under the PRB
list.
To find the number of ways six people could be arranged in six different chairs, you would
calculate six factorial (6!).
4
Graphing Calculator Manual
Type: 6 > MATH > PRB > 4: ! and press ENTER.
6! = 720.
Calculate 10! and 0!.
Combinations
The combination formula can also be used to compute dependent probabilities. The notation for
the number of combinations is nCr, where n is the total number of elements, and r is the number
being selected. Combinations are used when selecting a few elements from a larger number of
distinct elements.
Example: Ice Cream
An ice cream parlor offers 6 flavors of ice cream. Kristen would like to purchase 2 flavors of ice
cream. In how many ways can Kristen choose 2 flavors out of the 6 flavors?
In order to find the number of ways of choosing two flavors out of
six, we would need to calculate 6C2.
Type 6 > MATH > PRB > 3:
nCr and press ENTER.
Press the number 2 key and press
ENTER.
There are 15 different combinations of two flavors of ice cream.
Calculate 6C3 and 8C3 .
Permutations
The permutation formula can be used to compute dependent probabilities. The notation for the
number of permutations is nPr, where n is the total number of elements, and r is the number
being selected. Permutations are used when trying to find all possible arrangements of elements
taken from a larger selection. Arrangements involve putting the elements in a particular order.
Chapter 5: Discrete Random Variables and Their Probability Distributions
5
If Kristen’s story changes as below, then permutations apply
rather than combinations.
An ice cream parlor offers 6 flavors of ice cream. Kristen
would like to purchase 2 flavors of ice cream and concerned as
to which flavor is on the top and which flavor is on the bottom
(i.e. Kristen is concerned about the arrangement of the flavors).
In how many ways can Kristen arrange 2 flavors out of the 6
flavors?
In order to find the number of arrangements of choosing two flavors out of six, we would need to
calculate 6P2.
Type 6 > MATH > PRB > 2: nPr and press ENTER.
Press the number 2 key and press ENTER.
There are 30 different arrangements of two flavors of ice cream.
Calculate 6P3 and 8P3 .
Binomial Distribution
Randomly Generating Number of Successes From a Binomial Distribution
There are many situations in statistics where you need to
generate numbers from distributions where the numbers are not
equally likely to occur. One of the most commonly used
distributions used in statistics is the discrete Binomial
distribution.
The TI-84 Plus has a built-in function to generate random real
numbers from a specific Binomial distribution. The random real
number represents an x value, the number of successes.
Select MATH > PRB > 7:randBin(
Type 3, 0.3,5) and press ENTER.
The screen shot on the left repeated the Binomial experiment 5
times. Each time there were 3 trials with a probability of success
were 2, 1, 0, 1, and 2 respectively.
Generate 5 random numbers from a Binomial distribution with 3 trials and 0.9 probability of
success.
6
Graphing Calculator Manual
The syntax for the randBin( function is randBin(n, p, r). This will generate r random numbers
representing x the number of successes from a binomial distribution with n number of trials and p
probability of success on a given trial. Note: if r = 1, you may omit it.
Compute Binomial Probabilities
The command for computing a probability at x successes for a discrete Binomial distribution is
binompdf(.
To find the probability of x successes out of n trials, each with probability p of success, type
binompdf(n, p, x).
Example: VCR’s
Suppose that 5% of all VCR’s manufactured by an electronics
company are defective. Three VCR’s are selected at random. What
is the probability that exactly one of them is defective? P(x = 1)
Select 2nd > VARS (DISTR) > A:
binompdf( and press ENTER.
Type: 3, 0.05, 1) and press ENTER.
The result is 0.135375 or ≈ 13.5% chance that exactly one of
them is defective.
Calculate the same probability with 8 VCR’s selected at random, rather than 3.
Now there is ≈ 27.9% chance that exactly one of them is defective.
Compute Cumulative Binomial Probabilities
The command for the probability for a cumulative number of
successes from 0 to x for a discrete Binomial distribution with n
number of trials and p probability of success on any given single
trial is binomcdf( . P(number of successes ≤ x)
Using the same Binomial distribution of 3 VCR’s as above, what
is the probability that zero or one of them is defective? P(x ≤ 1)
Select 2nd > VARS (DISTR) > B: binomcdf( and press
ENTER.
Type: 3, 0.05, 1) and press ENTER.
Chapter 5: Discrete Random Variables and Their Probability Distributions
7
The result is 0.99275 or ≈ 99.3% chance that at most one of them is defective.
Calculate the same probability with 8 VCR’s selected at random, rather than 3.
Now there is ≈ 94.3% chance that at most one of them is defective.
Compute Poisson Probabilities
The command for computing the probability of x occurrences
within a given interval for a discrete Poisson distribution with a
mean number of occurrences λ is poissonpdf(λ, x).
P(number of occurrences = x)
Example: Telemarketing
Suppose that a household receives, on average, 9.5 telemarketing calls per week. Find the
probability that the household receives 6 calls this week.
Select 2nd > VARS (DISTR) > C: poissonpdf( and press
ENTER.
Type: 9.5, 6) and press ENTER.
The result is 0.076420796 ≈ 7.6% chance that the household
receives 6 calls this week.
Find the probability that the household receives 10 calls this week.
There is ≈ 12.4% chance that the household receives 10 calls this week.
Compute Cumulative Binomial Probabilities
The command for computing the probability of at most x
occurrences (cumulative) within a given interval for a discrete
Poisson distribution with a mean number of occurrences λ is
poissoncdf(λ, x). P(number of occurrences ≤ x)
Using the same Poisson distribution of Telemarketing calls as
above, what is the probability that the household receives at
most 6 calls this week? P(x ≤ 6)
8
Graphing Calculator Manual
Select 2nd > VARS (DISTR) > D: poissoncdf( and press
ENTER.
Type: 9.5, 6) and press ENTER.
There is ≈ 16.5% chance that the household receives at most 6
calls this week.
Find the probability that the household receives at most 10 calls this week.
There is ≈ 64.5% chance that the household receives at most 10 calls this week.
Geometric Probabilities
Your calculator can compute probabilities for a geometric random variable with probability of
success p using the geometpdf( command, located on the DISTR page. To find the probability of
the random variable taking the value x, type geometpdf(p, x).
Example: Car Ignition
Suppose that a car with a bad starter can be started 90% of the time by turning on the ignition.
What is the probability that it will take three tries to get the car started? Type geometpdf(0.9, 3);
the answer is 0.9%.
Cumulative Geometric Probabilities
As with the binomial and cumulative probability functions, there is a cumulative version
geometcdf( . It can be used to find the probability that a geometric random variable will take a
value of at most x by typing geometcdf(p, x).