Download Discrete Random Variables

6.1
Discrete Random Variables
Random Variables
A
random variable is a numeric measure
of the outcome of a probability
experiment



Random variables reflect measurements that
can change as the experiment is repeated
Random variables are denoted with capital
letters, typically using X (and Y and Z …)
Values are usually written with lower case letters,
typically using x (and y and z ...)
Examples (Random Variables)
●
Tossing four coins and counting the
number of heads


●
The number could be 0, 1, 2, 3, or 4
The number could change when we
toss another four coins
Measuring the heights of students

The heights could change from student
to student
Discrete Random Variable
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A discrete random variable is a random
variable that has either a finite or a countable
number of values


●
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A finite number of values such as {0, 1, 2, 3, and 4}
A countable number of values such as {1, 2, 3, …}
Discrete random variables are designed to
model discrete variables (see section 1.2)
Discrete random variables are often “counts of
…”
Example (Discrete Random
Variable)
●
The number of heads in tossing 3 coins
(a finite number of possible values)

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There are four possible values – 0 heads, 1
head, 2 heads, and 3 heads
A finite number of possible values – a
discrete random variable
This fits our general concept that discrete
random variables are often “counts of …”
Discrete Random Variables
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●
Other examples of discrete random variables
The possible rolls when rolling a pair of dice

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The number of pages in statistics textbooks

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A finite number of possible pairs, ranging from (1,1) to
(6,6)
A countable number of possible values
The number of visitors to the White House in a
day

A countable number of possible values
Continuous Random
Variable
●
A continuous random variable is a
random variable that has an infinite, and
more than countable, number of values

●
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The values are any number in an interval
Continuous random variables are
designed to model continuous variables
(see section 1.1)
Continuous random variables are often
“measurements of …”
Example (Continuous
Random Variable)
●
●
An example of a continuous random variable
The possible temperature in Chicago at noon
tomorrow, measured in degrees Fahrenheit



The possible values (assuming that we can measure
temperature to great accuracy) are in an interval
The interval may be something like (–20,110)
This fits our general concept that continuous
random variables are often “measurements of …”
Continuous Random Variables
●
●
Other examples of continuous random
variables
The height of a college student

●
The length of a country and western song

●
A value in an interval between 3 and 8 feet
A value in an interval between 1 and 15 minutes
The number of bytes of storage used on a
80 GB (80 billion bytes) hard drive

Although this is discrete, it is more reasonable to
model it as a continuous random variable
between 0 and 80 GB
Probability Distribution
●
●
The probability distribution of a
discrete random variable X relates
the values of X with their
corresponding probabilities
A distribution could be



In the form of a table
In the form of a graph
In the form of a mathematical formula
Probability Distribution
●
●
If X is a discrete random variable and x
is a possible value for X, then we write
P(x) as the probability that X is equal to
x
Examples


In tossing one coin, if X is the number of
heads, then P(0) = 0.5 and P(1) = 0.5
In rolling one die, if X is the number rolled,
then
P(1) = 1/6
Probability Distribution
 Properties
of P(x)
 Since P(x) form a probability distribution,
they must satisfy the rules of probability


 In
0 ≤ P(x) ≤ 1
Σ P(x) = 1
the second rule, the Σ sign means to
add up the P(x)’s for all the possible x’s
Probability Distribution

An example of a discrete probability
distribution

All of the P(x) values are positive and they
add up to 1
NOT a Probability Distribution
●
An example that is not a probability
distribution
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Two things are wrong


P(5) is negative
The P(x)’s do not add up to 1
Probability Histogram
A
probability histogram is a histogram
where


A
The horizontal axis corresponds to the
possible values of X (i.e. the x’s)
The vertical axis corresponds to the
probabilities for those values (i.e. the P(x)’s)
probability histogram is very similar to a
relative frequency histogram
Probability Histogram
 An
example of a probability histogram
 The
histogram is drawn so that the height
of the bar is the probability of that value
Mean of a Probability
Distribution
●
The mean of a probability distribution
can be thought of in this way:

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There are various possible values of a
discrete random variable
The values that have the higher
probabilities are the ones that occur more
often
The values that occur more often should
have a larger role in calculating the mean
The mean is the weighted average of the
values, weighted by the probabilities
Mean of a Discrete Random
Variable
●
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The mean of a discrete random
variable is
μX = Σ [ x • P(x) ]
In this formula



x are the possible values of X
P(x) is the probability that x occurs
Σ means to add up these terms for all
the possible values x
Mean
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Example of a calculation for the mean
[ x • P(x) ]
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Add: 0.2 + 1.2 + 0.5 + 0.6 = 2.5
The mean of this discrete random variable
is 2.5
Law of Large Numbers
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The mean can also be thought of
this way (as in the Law of Large
Numbers)

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If we repeat the experiment many times
If we record the result each time
If we calculate the mean of the results
(this is just a mean of a group of
numbers)
Then this mean of the results gets closer
and closer to the mean of the random
variable
Expected Value
●
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The expected value of a random variable
is another term for its mean
The term “expected value” illustrates the
long term nature of the experiments – as
we perform more and more experiments,
the mean of the results of those
experiments gets closer to the “expected
value” of the random variable
Variance
●
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The variance of a discrete random variable is
computed similarly as for the mean
The mean is the weighted sum of the values
μX = Σ [ x • P(x) ]
The variance is the weighted sum of the
squared differences from the mean
σX2 = Σ [ (x – μX)2 • P(x) ]
The standard deviation, as we’ve seen
before, is the square root of the variance …
σX = √ σX2
Variance
 The
variance formula
σX2 = Σ [ (x – μX)2 • P(x) ]
can involve calculations with many
decimals or fractions
 An equivalent formula is
σX2 = [ Σ x2 • P(x) ] – μX2
 This formula is often easier to compute
Good News!
 The
variance can be calculated by hand,
but the calculation is very tedious
 Whenever possible, use technology
(calculators, software programs, etc.) to
calculate variances and standard
deviations
Summary
 Discrete
random variables are measures
of outcomes that have discrete values
 Discrete random variables are specified
by their probability distributions
 The mean of a discrete random variable
can be interpreted as the long term
average of repeated independent
experiments
 The variance of a discrete random
variable measures its dispersion from its
mean
Let’s Try Some…
 Page
 Be
300 – 303
Calculator Ready!
Determine whether the random variable is
discrete or continuous. State the possible
values of the random variable.
a)
The amount of rain in Seattle during April.
b)
The number of fish caught during a fishing
tournament
c)
The number of customers arriving at a bank
between noon and 1pm
d)
The time required to download a file from the
internet
Determine whether the distribution
is a discrete probability
distribution.
X
P(x)
100
.1
200
.25
300
.2
400
.3
500
.1
In the following probability distribution, the random
variable X represents the number of activities a parent
of a K-5th grade student is involved in
X
0
1
2
3
P(x)
.035
.074
.197
.320
4
.374
a) Verify that this is a discrete probability distribution
b) Draw a probability histogram
X P(x)
0 .035
C) Compute and interpret the mean of the random
variable X.
1
2
3
4
D) Compute the variance of random variable X.
.074
.197
.320
.374
E) Compute the standard deviation of random variable x.
F) What is the probability that a randomly selected
student has a parent involved in 3 activities.
G) What is the probability that a randomly selected
student has a parent involved in 3 or 4 activities.
An investment counselor calls with a hot stock tip. He believes that if
the economy remains strong, the investment will result in a profit of
$50,000. If the economy grows at a moderated pace, the investment
will result in a profit of $10,000. however, if the economy goes into
recession, the investment will result in a loss of $50,000. You contact
an economist who believes that there is a 20% probability the
economy will remain strong, a 70% probability that the economy will
grow at a moderate pace, and a 10% probability that the economy
will slip into recession. What is the expected profit from this
investment.