Download 1 Random Variables 2 Probability Models

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Random Variables
De…nition 1 A random variable assumes a value based on the outcome of a
random event.
There are two types of random variables. Each type creates a very di¤erent
probability model with its own set of rules and computations.
De…nition 2 Discrete random variables can take one of a …nite number of
distinct outcomes. Discrete random variables jump from one state to the next
with nothing in between.
Example 1 The number of innings in a baseball game, number of players on
a basketball team and the number of cards in a deck are all discrete random
variables.
De…nition 3 Continuous random variables can take any numeric value
within a range of values.
Example 2 Height and weight are all continuous random variables.
Discrete probability models can use sample spaces to compute probabilities.
That does not work with continuous random variables which have an in…nite
number of values. The bell curve is an example of a continuous probability
model. In a continuous model, probability is computed as area under the
curve. As a general rule, such computations require calculus! Fortunately, the
normal curve is so important and well known there are many tools (TI-83/84,
Excel, SAS, etc.) to compute probabilities without resorting to calculus.
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Probability Models
De…nition 4 A probability model is a mathematical description of an experiment consisting of two parts: a disjoint listing of all outcomes of the experiment
and an assignment of probabilities to each outcome.
A probability model is similar to a sample space but can describe more
complex outcomes because there is no longer the requirement that outcomes be
equally likely. Recall that the sample space for rolling a pair of dice has 36
outcomes.
(1; 1) (2; 1) (3; 1) (4; 1) (5; 1) (6; 1)
(1; 2) (2; 2) (3; 2) (4; 2) (5; 2) (6; 2)
(1; 3) (2; 3) (3; 3) (4; 3) (5; 3) (6; 3)
(1; 4) (2; 4) (3; 4) (4; 4) (5; 4) (6; 4)
(1; 5) (2; 5) (3; 5) (4; 5) (5; 5) (6; 5)
(1; 6) (2; 6) (3; 6) (4; 6) (5; 6) (6; 6)
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Example 3 Construct the probability model for the sum of two dice.
sum x 2
3
4
5
6
7
1
2
3
4
5
6
P (x)
36
36
36
36
36
36
sum x 8
9
10 11 12
5
4
3
2
1
P (x)
36
36
36
36
36
Exercise 1 Construct the sample space for rolling one 3-sided die and one 4sided die. Construct the probability model for the sum of the two dice.
Probability models have two constraints. No probabilities may be less than
0 and the total of all the probabilities must be 1.
Problem 1 Is the following a probability model?
x
1
3
5
7
9
1
2
3
4
5
P (x) 10
10
10
10
10
Problem 2 Is the following a probability model?
x
1
3
5
7
9
2
3
4
3
P (x) 102 10
10
10
10
Problem 3 Is the following a probability model?
x
2
4
6
8
10
1
1
3
4
1
P (x) 10
10
10
10
10
Problem 4 What missing value will make the following a probability model?
x
1
3
5
7
9
1
2
3
??
2
P (x) 12
12
12
12
12
2
The expected value (or mean) of a probability model is given by
p(x).
x
=
X
x
Example 4 Compute the expected value for the probability model of the sum of
two dice.
sum x 2
3
4
5
6
7
1
2
3
4
5
6
P (x)
36
36
36
36
36
36
sum x 8
9
10 11 12
5
4
3
2
1
P (x)
36
36
36
36
36
5
36
So
+9
X
1
=
x p(x) = 2 36
+3
4
3
2
36 + 10
36 + 11
36 + 12
x
2
36 + 4
1
36 = 7.
3
36
+5
4
36
+6
5
36
+7
6
36
+8
Problem 5 Compute the expected value for the probability model given below.
x
-4 -2 1
2
3
2
1
5
1
1
P (x) 10
10
10
10
10
Problem 6 A box contains four slips of paper containing the numbers -2, -1,
0, 2. A game consists of randomly selecting a slip of paper from the box and
receiving that amount in dollars. You may play this game as many times as you
wish.
1. Will you play this game?
2. Compute the expected value.
3. If you play this game 100 times, what will the monetary result be?
Problem 7 A box contains four slips of paper containing the numbers -10, 1,
2, 2. A game consists of randomly selecting a slip of paper from the box and
receiving that amount in dollars. You may play this game as many times as you
wish.
1. Will you play this game?
2. Compute the expected value.
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Problem 8 Now the box contains fourteen million slips of paper. One slip
contains the value …ve million. All the other slips contain a -1. Do you play
this game?
Problem 9 You pay $1 to play a game. The game consists of rolling a pair of
dice. If you observe a sum of 7 or 11 you receive $4. If not, you receive nothing.
1. How is receiving $4 di¤ erent from winning $4?
2. What is the expected value of this game?
Problem 10 You draw a card from a deck.
you get an Ace you get $10.
If you get a club you get $5. If
1. Create a probability model for this game.
2. Compute expected value of a single play of this game
Problem 11 An investor buys a plot of land for $40,000 with hopes that land
values will go up with the arrival of a relocated MLB team. There is a 65%
chance that the land will appreciate to a value of 65,000. There is a 35% chance
the value of the land will fall to a value of 30,000. Compute the expected value
of this investment.
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Exercises
1. Navidi/Monk: 17-22, 27-32, 33-38 (compute mean only), 39, 40, 41 (not f.),
51, 52, 53, 55, 56, 57
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