Download Probability Distributions and Expected Value

You’re familiar with functions like
f (x) = x2 + 2x + 7
Probability Distributions
and Expected Value
where for every x value input you get a
unique f (x) value output. Normally, we
call the f (x) value the y value.
So for every value of the variable x, a
unique, well-defined value is assigned to
the variable y
A random variable is like the y variable
in our function, except its value can only
be determined by the outcome of an
experiment.
A random variable can be discrete or
continuous. A continuous random
variable can take on an infinite number of
continuous values, while discrete random
variables take on only a finite number of
values.
If you assign the average numerical
semester grade of a student chosen at
random to the value of a random variable,
that random variable is continuous. If you
choose a student at random and assign
grade points corresponding to the letter
grade to the random variable, that random
variable is discrete.
We will be dealing with discrete random
variables in this section.
Example: An instructor in a large
class curves his semester grades to
15% of the students receive A’s
and D’s; 30% receive B’s and C’s;
and 10% receive F’s.
Let x be the random variable that takes on
the value of the grade points earned of a
student chosen at random.
Since we know how this instructor
distributes his grade, we can easily
examine the distribution of probabilities
for each value of the random variable:
x
0
1
2
3
4
P(x)
.1
.15
.3
.3
.15
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This table represents the probability
distribution function for the random
variable, x.
Definition: A probability
distribution function of a discrete
random variable x is a table that
assigns the probability for each
possible value of x.
A probability histogram is a
way to picture a probability
distribution. It plots a bar
with the height of the
probability at each values of
the random variable, x.
Probability Histogram of Grade Points
Probability
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Classwork Example: Find the
probability function and draw
histogram for the random variable x
whose value is the sum of one roll of
2 fair dice.
.2
.1
0
1
2
3
Grade Points Earned
4
Definition: The expected value, E(x), of
a random variable that can take on n
values, x1, x2, . . ., xn, is calculated:
E(x) = x1P(x1) + x2P(x2) + . . . + xnP(xn)
the sum of the products of the values of the
variable times the probability of each of those
values.
Example: The expected value of the
variable assigned to be the grade points
earned by a student in that instructor’s
class is:
(0)(.1) + (1)(.15) + (2)(.3) + (3)(.3) +
(4)(.15) = 2.25
So if you take that instructor’s class,
you can expect to earn a C.
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Classwork Example (cont’d):
What value do you expect on the roll
of a pair of fair dice?
Binomial probability distributions: In
the last section we calculated the
probability that a basketball player that
hits 60% of his three throws will make 8
out of 12 free throws during a basketball
games.
If we assign the value of a random
variable x to the number of free throws out
of 12 that the player makes, we have the
probability distribution:
x
0
P(x) .00002
x
6
1
.0003
7
2
3
4
.0025 .0125
8
9
10
5
.042 .1009
11
Logically, how many free throws out of
12 do we expect our player to throw?
Does this match the expected value that
we would calculate using our formula?
12
P(x) .1766 .227 .213 .1419 .0639 .0174 .0022
Expected value for binomial
probability: E (x) = np
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