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Random Variables and Probability Distributions
Some Definitions
Random Variable is a function that assigns numerical values to sample points (one
and only value to each sample point) of the sample space of an experiment. Usual
notations are X, Y , Z, etc. A random variable is discrete if it can assume only a
countable number of values, and continuous if it can assume any value in one or more
intervals.
Probability distribution of a discrete random variable is a graph or a table or a formula that specifies both the values of the variable and the corresponding probabilities.
Usually p(x) is used to denote the probability that the random variable X takes the
P
value x. Also, note that p(x) ≥ 0 for all x and
p(x) = 1.
Mean (also called expected value) and variance of probability distributions of a
discrete random variable:
Mean = µ = Σxp(x)
Variance = σ 2 = Σx2 p(x) − µ2
Example. Toss a fair coin twice. Let X be the number of heads observed. Then X
is a discrete random variable that takes the values 0, 1, 2 with probabilities 1/4, 1/2,
and 1/4, respectively. This probability distribution may be displayed as follows.
Probability distribution of X (the number of heads)
X
p(x)
0
1/4
1
1/2
2
1/4
Mean and variance:
µ = Σxp(x) = 1
1
Variance = σ 2 = Σx2 p(x) − µ2 = 1/2
The above probability distribution can also be displayed as follows:
p(x) =
2!
1
1
( )x ( )2−x ,
x!(2 − x)! 2
2
x = 0, 1, 2
Example.
The number of training units that must be passed before a complex computer
software is mastered varies from one to five depending on the student. After much
experience, the software manufacturer has determined the following probability
distribution that describes the fraction of users mastering the software after each
number of training units.
Number of Units :
Probability of Mastery:
1
2
3
4
5
.1
.25
.4
.15
.1
a) Calculate the mean number of units necessary to master the program. (µ = 2.9)
b) If the firm wants to ensure that at least 75% of the students master the program,
what is the minimum number of training units that must be administered? (3)
Example
The Made Fresh Daily Apple Pie Company knows that the number of pies sold
each day varies from day to day. The owner believes that on 50% of the days she
sells 100 pies. On another 25% of the days she sells 150 pies, and she sells 200
pies on the remaining 25% of the days. To make sure she has enough product,
the owner bakes 200 pies each day at a cost of $2 each. Assume any pies that go
unsold are thrown out at the end of the day. If she sells the pies for $4 each, find
the probability distribution for her daily profit. Also, find her expected profit.
(profit (x): 0 200 400
prob(x):
0.5 0.25 0.25
Expected profit: µ = 150 )
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Example: Expected Lotto Winnings.
The chance of winning a Lotto game is 1 in 23 million. Suppose you buy a $1
Lotto ticket in anticipation of winning the $7 million grand prize. Calculate your
expected net winnings for this single ticket. (Expected profit = -0.70)
Example.
The weather at a Caribbean island is perfect for a tourist destination. The chance
that it rains on any particular day is only 20% and each day’s weather is independent of every other day’s weather. Suppose you plan a vacation at this island
for three days. Let X be the number of rainy days. Write down the probability
distribution of X.
x:
0
1
2
3
p(x) 0.512 0.384 0.096 0.008
What is the probability that you get rain on two days? one day? no day? at least two
days?
(0.096; 0.384; 0.512; 0.104)
Binomial Experiment is consist of n ≥ 1 independent and identical trials where each
trial has two possible outcomes S (for Success) and F (for Failure) such that P (S) = p
is the same for each of the n trials. In a binomial experiment, the discrete variable
X (= number of success) is called the binomial random variable and its probability
distribution given below is called binomial probability distribution.
p(x) =
n!
px (1 − p)n−x ,
x!(n − x)!
mean =µ = np, variance = σ 2 = np(1 − p).
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x = 0, 1, 2, . . . , n.
Example.
The weather at a Caribbean island is perfect for a tourist destination. The chance
that it rains on any particular day is only 20% and each day’s weather is independent of every other day’s weather. Suppose you plan a vacation at this island for
ten days. What is the probability that you get rain on two days? one day? no
day? at least three days?
(Ans. 0.302; 0.269; 0.107; 0.322)
Example
The Statistical Abstract of the United States reports that 30% of the country’s
households are composed of one person. If 20 randomly selected homes are to
participate in a Nielson survey to determine television ratings, find the probability
that fewer than five of these homes are one-person households. (0.238)
Example.
Investing is a game of chance. Suppose there is a 40% chance that a risky stock
investment will end up in a total loss of your investment. Because the rewards
are so high, you decide to invest in ten independent risky stocks. What is the
probability that at least two of these ten stocks end up in total losses? (0.954)
Example.
The probability that a person responds to a mailed questionnaire is 0.4. What is
the probability that of 25 questionnaires, more than 15 will be returned? (0.013)
Example.
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The median time a patient waits to see a doctor in a large clinic is 20 minute. On
a day when 25 patients visit the clinic, what is the probability that more than half
but fewer than 18 will wait more than 20 minutes? (0.478)
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*** This section is intended for Honors Section only ***
Poisson Probability Distribution
The Poisson random variable X is the observed number of rare events during a unit
of measurement (e.g., time, area, volume, weight, distance, etc.) and its probability
distribution is given by p(x) =
λx e−λ
x! , x
= 0, 1, . . . , ; where λ is the expected number
of events during the given unit of measurement. For this distribution, both mean and
variance are equal to λ (i.e. µ = σ 2 = λ)
Examples of some events and units:
Number of accidents (event) per month (unit)
Number of cancer deaths (event) per year (unit)
Number of diseased trees (event) per acre (unit)
Number of airline fatalities (event) per month (unit)
Number of hurricanes (event) per season (unit)
Number of misprints (event) per page (unit) of a book
Example:
The number of hurricanes to threaten the USA during a hurricane season follows a
Poisson distribution with a mean of 6.0. Using Poisson distribution, find the probability that at least four hurricanes will threaten USA during the next hurricane
season.
Example:
Historical data suggest that an average of five UCF instructors miss their Monday
morning classes. Using Poisson distribution, find the probability that at least five
UCF instructors will not show up for their next Monday morning classes.
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Geometric Probability Distribution
The binomial random variable is obtained by observing a sequence of n independent
Bernoulli trials and counting the number of successes in n trials. Suppose now that we
do not fix the number of Bernoulli trials in advance but instead continue to observe
the sequence of Bernoulli trials until we observe a success. The random variable X of
interest is the number of trials (or equivalently, the number of failures) needed before
the first success. The probability distribution of x is called geometric distribution and
is given by
p(x) = p(1 − p)x , x = 0, 1, . . . , where p is the probability of success.
1−p
p
1−p
2
σ = p2
µ=
Example
A couple decides that they will have kids until a girl is born. the outcome of
each birth is independent event, and the probability that a girl will be born is
1/2. What is the probability that they will have four kids? What is the expected
number of kids per family? (= µ + 1 since µ is the expected number of boys before
a girl is born)
Example
A student decides to continue taking STA 2023H class until a B or better grade
is earned. Assuming independent outcomes and 40% chance of passing with a B
or better grade in each try, calculate the probability that the student will have to
take this class four times.
Additional Practice Problems (with answers in parenthesis): 4:18 (.65;.75;.85;.95;.70;.65);
4:28 (.15; .15); 4:56 (.121; .034; .081); 4:64 (.064; .936; µ = 30; σ = .8485); 4:90 (.010;
yes); 4:122 (.23; .143; .10); 4:128 ( .917; .033);
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