Discrete probability distributions

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Discrete probability distributions

• 3)For every possible x value,
• 0 lt P(x) lt 1.
• 4) For all values of x,
• S P(x) 1.

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• Think About It

• In a game of dice a friend gives you a choice. If
  an even number is rolled you win 100, if a 5 is
  rolled you win 100. Which option would you
  choose? Why?
• You friend now states if a 5 is rolled you will
  win 200. Which option would you choose? Why?
• Again, your friend increased a roll of 5 to 300.
• Which option would you choose? Why
• Finally your friend states a roll of 5 will win
  you 400. Which option would you choose? Why?

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Random Variable -

• A numerical variable whose value depends on the
  outcome of a chance experiment

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Two types

• Discrete count of some random variable
• Continuous measure of some random variable

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Random Variable
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Discrete

• The probabilities pi must satisfy two
  requirements
• Every probability pi is a number between 0 and 1.
• 0 lt P(x) lt 1.
• The sum of the probabilities is 1.
• S P(x) 1.
• To find the probability of any event, add the
  probabilities pi of the particular values xi that
  make up the event.

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Discrete Probability Distribution

• A distribution of a random variable gives its
  possible values and their probabilities.
• Usually displayed in a table, but can be
  displayed with a histogram or formula

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Probability Distribution
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What are the chances

• What is P(X gt 2)
• Show that this is a legitimate probability
  distribution.

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Let x be the number of courses for which a
randomly selected student at a certain university
is registered. X 1 2 3 4 5 6 7
P(X) .02 .03 .09 ? .40 .16 .05 P(x 4) P(x
lt 4) P(x lt 4) What is the probability that
the student is registered for at least five
courses?
Why does this not start at zero?
.25
.14
P(x gt 5) .61
.39
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• Example Babies Health at Birth
• Read the example on page 343.
• Show that the probability distribution for X is
  legitimate.
• Make a histogram of the probability distribution.
  Describe what you see.
• Apgar scores of 7 or higher indicate a healthy
  baby. What is P(X 7)?

Value 0 1 2 3 4 5 6 7 8 9 10
Probability 0.001 0.006 0.007 0.008 0.012 0.020 0.038 0.099 0.319 0.437 0.053
(a) All probabilities are between 0 and 1 and
they add up to 1. This is a legitimate
probability distribution.
(c) P(X 7) .908 Wed have a 91 chance of
randomly choosing a healthy baby.
(b) The left-skewed shape of the distribution
suggests a randomly selected newborn will have an
Apgar score at the high end of the scale. There
is a small chance of getting a baby with a score
of 5 or lower.
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Formulas for mean variance
Found on formula card!
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Dice
Expected Value Comparisons
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Tebow Time!
The NFL Draft is an annual event which is the
most common source of player recruitment. In the
first round of the 2010 NFL draft the Denver
Broncos selected Tim Tebow. At the position of
Quarterback Tebows ability was highly debated on
a national level. The Broncos Franchise took a
major risk, however, do no think for a second
this was not a calculated risk.
Imagine you are on the Broncos Management.
Judging by his record in College, analysts
predict Tebow has a 10 chance of becoming an
elite quarterback, pulling in 20 million for the
franchise. He has a 40 chance of being average,
bringing in 10 million. Otherwise, he will be
2nd or 3rd string which brings in no money and
would be a loss (the cost of the contract) of
9.7 million.
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• Example Apgar Scores Whats Typical?
• Consider the random variable X Apgar Score
• Compute the mean of the random variable X and
  interpret it in context.

Value 0 1 2 3 4 5 6 7 8 9 10
Probability 0.001 0.006 0.007 0.008 0.012 0.020 0.038 0.099 0.319 0.437 0.053
The mean Apgar score of a randomly selected
newborn is 8.128. This is the long-term average
Agar score of many, many randomly chosen
babies. Note The expected value does not need
to be a possible value of X or an integer! It is
a long-term average over many repetitions.
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• 1. A college instructor teaching a large class
  traditionally gives 10 As, 20 Bs, 45 Cs,
  15 Ds, and 10 Fs. If a student is chosen at
  random from the class, the students grade on a
  4-point scale (A 4) is a random variable X.
  Create the probability distribution of X.
• What is the probability that a student has a
  grade point of 3 or better in this class?
• Draw a probability histogram to picture the
  probability distribution of the random variable
  X.
• 2. Put all the letters of the alphabet in a hat.
  If you choose a consonant, I pay you 1. If you
  choose a vowel, I pay you 5. X is the random
  variable representing the outcome of the
  experiment.
• Create the distribution of X
• What is your expected payoff (value) in this
  game?

• You Try

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• 2. Put all the letters of the alphabet in a hat.
  If you choose a consonant, I pay you 1. If you
  choose a vowel, I pay you 5. X is the random
  variable representing the outcome of the
  experiment.
• What is the games variance? The Standard
  deviation?

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Let x be the number of courses for which a
randomly selected student at a certain university
is registered. X 1 2 3 4 5 6 7 P(X) .02 .03 .09 .2
5 .40 .16 .05 What is the expected value and
standard deviations of this distribution?
m 4.66 s 1.2018
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Is the formula the only way?!?!?!?!!?
Stat, 1Edit L1 Random Variable ( X ) L2
Probability (pi) Stat, Calc, 1 1-Var Stats 2nd
Stat L1 2nd Stat L2 1-Var Stats L1, L2
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Let x be the number of courses for which a
randomly selected student at a certain university
is registered. X 1 2 3 4 5 6 7 P(X) .02 .03 .09 .2
5 .40 .16 .05 What is the expected value and
standard deviations of this distribution?
m 4.66 s 1.2018
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• Box of 20 DVDs, 4 are defective. Select two from
  the box without replacement
• Identify your random variables.
• Create a Probability Distribution
• What is the mean (expected value) of the discrete
  random variable?
• What is the variance? The Standard Deviation?

.
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• Cars in a Town
• X number of vehicles owned by a household in a
  random town
• P(0) .05, P(1) .45, P(2) .275, P(3) .1,
  P(4) .075,
• P(5) .05
• Identify your random variables.
• Create a Probability Distribution
• What is the mean (expected value) of the discrete
  random variable?
• What is the variance? The Standard Deviation?

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• Book Editor
• X of errors that appear on a randomly
  selected page of a book
• X 0, 1, 2, 3 ,4
• P(0) .73, P(1) .16, P(2) .06, P(3) .04,
  P(4) .01
• Identify your random variables.
• Create a Probability Distribution
• What is the mean (expected value) of the discrete
  random variable?
• What is the variance? The Standard Deviation?

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• Flights from LA to Chicago
• X of flights that are on time out of 3
  independent flights
• P(0) .064, P(1) .288, P(2) .432, P(3) .216
• Identify your random variables.
• Create a Probability Distribution
• What is the mean (expected value) of the discrete
  random variable?
• What is the variance? The Standard Deviation?

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Linear combinations
Just add or subtract the means!
If independent, always add the variances!
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A nationwide standardized exam consists of a
multiple choice section and a free response
section. For each section, the mean and standard
deviation are reported to be mean SD MC 38
6 FR 30 7 If the test score is computed by
adding the multiple choice and free response,
then what is the mean and standard deviation of
the test?
m 68 s 9.2195
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Linear function of a random variable
The mean is changed by addition multiplication!

• If x is a random variable and a and b are
  numerical constants, then the random variable y
  is defined by
• and

The standard deviation is ONLY changed by
multiplication!
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Let x be the number of gallons required to fill a
propane tank. Suppose that the mean and standard
deviation is 318 gal. and 42 gal., respectively.
The company is considering the pricing model of a
service charge of 50 plus 1.80 per gallon. Let
y be the random variable of the amount billed.
What is the mean and standard deviation for the
amount billed?
m 622.40 s 75.60