Download Section 8.1 - Distributions of Random Variables • Definition: A

Section 8.1 - Distributions of Random Variables
• Definition: A random variable is a rule that assigns a number to each
outcome of an experiment.
• Example 1: Suppose we toss a coin three times. Then we could define the
random variable X to represent the number of times we get tails.
• Example 2: Suppose we roll a die until a 5 is facing up. Then we could define
the random variable Y to represent the number of times we rolled the die.
• Example 3: Suppose a flashlight is left on until the battery runs out. Then
we could define the random variable Z to represent the amount of time that
passed.
• Types of Random Variables
1. A finite discrete random variable is one which can only take on a limited
number of values that can be listed.
2. A infinite discrete random variable is one which can take on an unlimited
number of values that can be listed in some sort of sequence.
3. A continuous random variable is one which takes on an infinite number
of possible values. (i.e. usually measurements)
• Example 4: Referring to Example 1, find the probability distribution of the
random variable X.
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• Example 5: The number of cars waiting in line at the beginning of 2-minute
intervals at a certain bank was observed. The following data was collected:
Number of cars
0 1 2 3 4
Frequency of occurence 2 2 16 8 2
Let X represent the number of cars observed waiting in line and find the
probability distribution of X
• We use histograms to represent the probability distributions of random variables. We place the possible values for the random variable X on the horizontal
axis. We then center a bar around each x value and let its height be equal to
the probability of that x value.
• Example 6: Referring to Example 5,
a) Draw the histogram for the random variable X
b) Find P (X ≥ 2)
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Section 8.2 - Expected Value
• Example 1: Records kept by the cheif dietitian at the university caferteria
over a 25-wk period show the following weekly consumption of milk (in gallons):
Milk 200 201 202 203 204
Weeks 4
6
8
5
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Find the average number of gallons of milk consumed per week in the cafeteria.
• Definition: Let X denote a random variable that assumes the values x1 , x2 , . . . , xn ,
with associated probabilites p1 , p2 , . . . , pn , respectively. Then the expected value
of X, E(X), is given by:
E(X) = x1 p1 + x2 p2 + · · · xn pn
• Example 2: Referring to Example 1, let the random variable X denote the
number of gallons of milk consumed in a week at the cafeteria.
a) Find the probability distribution of X.
b) Compute E(X)
c) Draw a histogram for the random variable X
• Note: If we think of placing the histogram on a see-saw, the expected value
occurs where we would put the fulcrum to balance it.
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• Example 3: In a lottery, 3000 tickets are sold for $1 each. One first prize of
$5000, 1 second prize of $1000, 3 third prizes of $100, and 10 consolation prizes
of $10 are to be awarded. What are the expected net earnings of a person who
buys one ticket?
• Definition: A fair game means that the expected value of the player’s net
winnings is zero. (E(X) = 0)
• Example 4: Mike and Bill play a card game with a standard deck of 52 cards.
Mike selects a card from a well-shuffled deck and receives A dollars from Bill if
the card selected is a diamond; otherwise, Mike pays Bill a dollar. Determine
the value of A if the game is to be fair.
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• Definition: If P (E) is the probability of an event E occuring, then
a) The odds in favor of E occuring are
P (E)
P (E)
=
1 − P (E) P (E c )
[P (E) 6= 1]
b) The odds against E occuring are
1 − P (E) P (E c )
=
P (E)
P (E)
[P (E) 6= 0]
• Example 5: The probability of an event E occurring is .8.
a) What are the odds in favor of E occurring?
b) What are the odds against E occurring?
• Definition: If the odds in favor of an event E occuring are a to b, then the
probability of E occurring is
a
P (E) =
a+b
• Example 6: If a sports forecaster states that the odds of a certain boxer
winning a match are 4 to 3, what is the probability that the boxer will win the
match?
• Definition: The median is the middle value in a set of data arranged in
increasing or decreasing order (when there is an odd number of entries). If
there is an even number of entries, the median is the average of the two middle
numbers.
• Definition The mode is the value that occurs most frequently in a set of data.
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Section 8.3 - Variance and Standard Deviation
• Example 1: Draw the histograms for the random variables X and Y that have
the following probability distributions:
y P (Y = y)
1
.3
2
.1
3
.2
4
.1
5
.3
x P (X = x)
1
.1
2
.1
3
.6
4
.1
5
.1
• Definition: Suppose a random variable X has the following probability distribution:
x P (X = x)
x1
p1
x2
p2
·
·
·
·
·
·
xn
pn
and expected value E(X) = µ. Then the variance of the random variable X
is Var(X)=p1 (x1 − µ)2 + p2 (x2 − µ)2 + · · · + pn (xn − µ)2 .
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• Definition: The standard deviation of a random variable X, σ, is defined
by:
p
σ = V ar(X)
• Finding expected value and standard deviation using the calculator:
a) Enter the x-values into L1 and the corresponding probabilities into L2
(STAT→1:Edit)
b) On the homescreen type 1-Var Stats L1 , L2 (STAT→CALC→1:1-Var Stats)
c) x̄ is the expected value (mean)
d) σx is the standard deviation
e) To find variance, you would need to recall that V ar(X) = σ 2
• Example 2: Referring to Example 1,
a) Find the variance and standard deviation of X.
b) Find the variance and standard deviation of Y .
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• Chebychev’s Inequality: Let X be a random variable with expected value
µ and standard deviation σ. Then, the probability that a randomly chosen
outcome of the experiment lies between µ − kσ and µ + kσ is at least 1 − k12 ;
that is:
1
P (µ − kσ ≤ X ≤ µ + kσ) ≥ 1 − 2
k
• Example 4: A probability distribution has a mean of 10 and a standard deviation of 1.5. Use Chebychev’s inequality to estimate the probability that an
outcome of the experiment lies between 7 and 13
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Section 8.4 - The Binomial Distribution
• Binomial Experiments have the following properties:
1. The number of trials in the experiment is fixed.
2. There are 2 possible outcomes in the experiment: ”success” and ”failure”
3. The probability of success in each trial is the same.
4. Trials are independent of each other.
• Calculating a Binomial Probability:
1. Determine if the experiment is binomial.
2. Determine the number of trials (n).
3. Define ”success” in the experiment and determine the probability of that
success occuring (p).
4. Determine the number of ”successes” desired (r).
5. Calculate the desired probability:
(a) By hand, the probability is found by doing the following calculation:
P (X = r) = C(n, r)pr (1 − p)(n−r)
(b) We can use the calculator program BINOM to do these calculations for
us.
• Example 1: What is the probability of exactly 3 successes in 6 trials of a
binomial experiment in which p = 1/2?
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• Example 2: A fair die is cast 4 times. Compute the probability of obtaining
exactly one 6 in the four throws.
• Example 3: A biology quiz consists of 8 multiple choice questions. Five must
be answered correctly to receive a passing grade. If each question has 5 possible
answers of which only one is correct, what is the probability that a student who
guesses at random on each question will pass the examination?
• Definition: For a binomial random variable X, we have the following:
E(X) = np
Var(X) = np(1 − p)
p
σ = np(1 − p)
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• Example 4: At a certain university the probability that an entering freshman
will graduate within 4 years is .6. From an incoming class of 2000 freshman,
find
a) The expected number of students who will graduate within 4 years.
b) The standard deviation of the number of students who will graduate within
4 years.
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Section 8.5 - The Normal Distribution
• Up until now, we have been dealing with finite discrete random variables. In
finding the probability distribution, we could list the possible values in a table
and represent it with a histogram.
• Definition: For a continuous random variable, a probability density function
is defined to represent the probability distribution.
• Example 1:
• Note that the for a continous random variable, X, P (X ≤ x) = P (X < x)
• Definition: We concentrate on a special class of continuous probability distributions known as normal distributions. Each normal distribution is defined
by µ and σ. Each normal distribution has the following characteristics:
1.
2.
3.
4.
The
The
The
The
area under the curve is always 1.
curve never crosses the x−axis.
peak occurs directly above µ
curve is symmetric about a vertical line passing through the mean.
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• Example 2:
• Definition: The standard normal variable usually denoted by Z has a
normal probability distribution with µ = 0 and σ = 1.
• Example 3: Find and sketch the following:
a) P (Z ≤ 1.79)
b) P (Z ≥ 3.49)
c) P (−2 ≤ Z ≤ 1.79)
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• Suppose the random variable X has a normal distribution with mean µ and
standard deviation σ. We can calculate probabilities for X by converting the
normal distribution to a standard normal distribution. We do this by converting the bounds of X to bounds of Z using the following formula:
z=
x−µ
σ
• Example 4: Suppose X is a normal random variable with µ = 500 and σ = 75.
Find P (400 ≤ X ≤ 600)
• Example 5: Let Z be the standard normal variable. Find the values of z if z
satisfies:
a) P (Z ≤ z) = .8907
b) P (−z ≤ Z ≤ z) = .7820
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Section 8.6 - Applications of the Normal Distribution
• Example 1: According to the data released by the Chamber of Commerce
of a certain city, the weekly wages of factory workers are normally distributed
with a mean of $600 and a standard deviation of $50. What is the probability
that a worker selected at random from the city makes a weekly wage
a) of less than $600?
b) of more than $760?
c) between $550 and $650?
• Example 2: The scores on an Econ exam were normally distributed with a
mean of 72 and a standard deviation of 16. If the instructor assigns a grade of
A to 15% of the class, what is the lowest score a student may have and still
obtain an A?
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• Example 3: Let’s look at the following example. Suppose a fair coin is tossed
5 times. Let X represent the number of times the coin lands on heads. X has
a binomial distribution.
a) Find the probability distribution for X.
b) Draw the histogram for X.
c) Find P (X ≤ 3)
d) Use the appropriate normal distribution to approximate the binomial probability in part c.
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• Steps to approximating binomial probabilities with normal distributions:
1. Identify n and p.
2. Calculate µ and σ:
µ = np
p
σ = np(1 − p)
3. Convert the binomial random variable X to the normal random variable Y :
– Draw a rough sketch of the histogram of X and the normal curve of Y .
– Determine which rectangles you are wanting to include.
– Find the appropriate interval for the area under the normal curve by
adding or subtracting 0.5 to your bounds.
4. Compute the desired area under the normal curve using normalcdf.
• Example 4: A basketball player has a 60% chance of making a free throw.
What is the probability of her making 250 or more free throws in 400 trials? Use
the appropriate normal distribution to approximate the binomial distribution.
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