Random Variables
- A random variable \(X\) is a numerical measure of the outcome
of a probability experiment, so its value is determined by chance.
Example 1 Flip a coin three times. Let
\(X\) be the number of heads.
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| Outcomes |
HHH |
HHT |
HTH |
THH |
HTT |
THT |
TTH |
TTT |
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| X |
3 |
2 |
2 |
2 |
1 |
1 |
1 |
0 |
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Ross 5.1 The National Basketball
Association (NBA) draft lottery involves the 14 teams that had the worst
won-lost records during the preceding year. Fourteen pingpong balls,
numbered 1 through 14, are placed in an urn. From this urn, 4 balls will
be randomly selected, and the first pick in the draft of players about
to enter the league is awarded based on which set of balls is
chosen.
- How many possible choices of a set of 4 balls from the 14 in the urn
are there?
- Before selecting the balls, 250 of these possible outcomes are
assigned to the team with the worst last year record. How likely will
the team get the first pick?
- How about other teams?
| Rank (Worst to Best) |
Number of Sets Assigned |
| 1 |
250 |
| 2 |
199 |
| 3 |
156 |
| 4 |
119 |
| 5 |
88 |
| 6 |
63 |
| 7 |
36 |
| 8 |
28 |
| 9 |
17 |
| 10 |
11 |
| 11 |
8 |
| 12 |
7 |
| 13 |
6 |
| 14 |
5 |
Probability Distribution
- A probability distribution is a graph,
table, or formula \(P(x)\) that gives
the probability for each value of the random variable. A distribution is
discrete if the random variable \(X\) is discrete.
Let \(P(x)\) be the probability of
the discrete random variable \(X\)
takes the value of \(x\):
- \(0 \le P(x) \le 1\) for all \(x\).
- \(\sum P(x)=1\).
Example 2 Display the discrete probability
distribution when we roll a fair dice and record the number on the
dice.
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\(X\)
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5
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6
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\(P(X)\)
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Example 3 Display the discrete probability
distribution when we roll a pair of fair dice and record the sum of the
pair.
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\(X\)
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2
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3
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4
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5
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6
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12
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\(P(X)\)
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Example 4 Display the discrete probability
distribution when we roll a customized dice and record the number on the
dice.
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\(X\)
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2
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3
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4
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5
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6
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\(P(X)\)
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Example 5 Display the discrete probability
distribution when we roll a pair of the customized dice that we made in
Example 3 and record the sum of the pair.
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\(X\)
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2
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3
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\(P(X)\)
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Ross 5.4 A saleswoman has scheduled two
appointments to sell encyclopedias. She feels that her first appointment
will lead to a sale with probability 0.3. She also feels that the second
will lead to a sale with probability 0.6 and that the results from the
two appointments are independent. What is the probability distribution
of \(X\), the number of sales made?
Expected Values
The expected value of a discrete random
variable \(X\) whose possible values
are \(x_1, x_2, \cdots, x_n\), is
denoted by \(E[X]\) and is defined
by
\[E[x] = \sum_{i=1}^n x_i
P(x_i),\]
which represents, on average, what value one would expect to occur.
It is also called the mean of the random
variable or the weighted
average.
Example 6 Let \(X\) be the number facing up when we roll a
fair dice. Find \(E[X]\).
Example 7 Let \(X\) be the number facing up when we roll
the customized dice we created in Example 4. Find \(E[X]\).
Example 8 Suppose, in a casino, people bet
on a fair dice. They get $1 if the dice lands on a 3. What is the
minimum price that the casino should charge each person who plays in
this game, so that the casino will not lose money in the long run?
Example 9 In the same casino, people bet on
the sum of the two customized dice we created in Example 5. Create the
game prizes and find out the minimum price that the casino should charge
for each person who plays in the game you created.
Properties of expected values:
\[E[X+Y+Z] = E[X] + E[Y] +
E[Z]\]
Ross 5.9 The following are the annual
incomes of 7 men and 7 women who are residents of a certain community
(all numbers are in $1000). Suppose that a woman and a man are randomly
chosen. Find the expected value of the sum of their incomes.
| Men |
Women |
| 33.5 |
24.2 |
| 25.0 |
19.5 |
| 28.6 |
27.4 |
| 41.0 |
28.6 |
| 30.5 |
32.2 |
| 29.6 |
22.4 |
| 32.8 |
21.6 |
Ross 5.10 The following table lists the
number of civilian full-time law enforcement employees in eight cities.
Suppose that two of the cities are to be randomly chosen and all the
civilian law enforcement employees of these cities are to be
interviewed. Find the expected number of people who will be
interviewed.
| City |
Civilian law enforcement employees |
| A |
105 |
| B |
155 |
| C |
149 |
| D |
195 |
| E |
290 |
| F |
357 |
| G |
246 |
| H |
178 |
Ross 5.11 A building contractor has sent in
bids for three jobs. If the contractor obtains these jobs, they will
yield respective profits of 20, 25, and 40 (in units of $1000). On the
other hand, for each job the contractor does not win, he will incur a
loss (due to time and money already spent in making the bid) of 2. If
the probabilities that the contractor will get these jobs are,
respectively, 0.3, 0.6, and 0.2, what is the expected total profit?
Standard Deviation
The standard deviation of a discrete random variable \(X\) is
\[\sigma_X = \sqrt{\sum (x-\mu_X)^2\cdot
P(X)}\]
Example 10 Find the standard deviation of
the discrete random variable \(X\).
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\(X\)
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0
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1
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2
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3
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\(P(X)\)
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0.01
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0.10
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0.38
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0.51
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x <- c(0, 1, 2, 3)
p <- c(0.01, 0.10, 0.38, 0.51)
mu <- sum(x * p)
sqrt(sum((x - mu)^2 * p))