Download 3.1 Random Variable (continue) Example 1. A bag contains 6 green

3.1 Random Variable (continue)
Example 1. A bag contains 6 green marbles and 4 red marbles. We randomly select a marble in the
bag, observe the color, and then put it back to the bag. This procedure is done 6 times. Let the random
variable X be the number of times when a green marble is observed.
a) What are possible values of X?
b) Build the probability distribution table for X?
Given a sequence of n Bernoulli trials with the probability of success p and the probability of failure q,
the binomial distribution is given by
P (X = k) = C(n, k)pk q n−k
for k = 0, · · · , n.
3.2 Measure of Central Tendency
The average or mean of the n numbers x1 , x2 , · · · , xn , denoted by µ or x̄, is given by
µ=
x1 + x2 + · · · + xn
.
n
The median of a set of numerical data is the middle number when the numbers are arranged in order
of size and there is an odd number of entries in the set. In the case that the number of entries in the
set is even, the median is the mean of the two middle numbers.
The mode of a set of observations is the observation that occurs more frequently than the others. If
the frequency of occurrence of two observations is the same and also greater than the frequency of
occurrence of all the other observations, then we say the set is bimodal and has two modes. If no one
or two observations occurs more frequently than the others, we say that the set has no mode.
Example 2. What are the mean, median, and mode of the set of numbers 1, 4, 4, 5, 9, 9, 9?
What are the mean, median, and mode of the set of numbers 1 ,1 ,4 ,5, 5, 9?
Finding Mean and Median Using Your Calculator Given a List of Data:
(1) Press STAT.
(2) Select EDIT.
(3) In the first column, enter the values.
(4) Exit to the home screen.
(5) Press STAT.
(6) Scroll to the right to CALC.
(7) Select 1-Var Stats (Option 1)
(8) Press L1 (2nd → 1).
(9) Press ENTER.
(10) The mean is given by x̄. To find the median, scroll down until you see Med.
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Example 3. The daily maximum temperature in degrees Fahrenheit for the month of October in College
Station follow:
85, 82, 85, 90, 89, 77, 63, 70, 85, 89,
89, 89, 86, 87, 82, 70, 90, 79, 82, 86, 90,
88, 86, 87, 88, 62, 63, 67, 71, 75, 84
Find the mean temperature and the median temperature.
“Mean” is not always the best representation of what outcomes are occurring “most of the time” because
it is sensitive to extreme values.
Example 4. What are the mean, median, and mode of the set of numbers 1 ,1 ,4 ,5, 5, 9, 735?
Finding Mean and Median Using Your Calculator Given the Random Variable and the
Frequency or Probability:
(1) Press STAT.
(2) Select EDIT.
(3) In the first column, enter the values the random variable X takes on.
(4) In the second column, enter the frequency or probability for each value of X.
(5) Exit to the home screen.
(6) Press STAT.
(7) Scroll to the right to CALC.
(8) Select 1-Var Stats (Option 1)
(9) Press L1 (2nd → 1). Press the comma key. Press L2 (2nd → 2).
(10) Press ENTER.
(11) The mean is given by x̄. To find the median, scroll down until you see Med.
Let X denote the random variable that has values x1 , x2 , · · · , xn , and let the associated probabilities be
p1 , p2 , · · · , pn . The expected value or mean of the random variable X, denoted by E(X) or µ, is
E(X) = x1 p1 + x2 p2 + · · · + xn pn .
Note: If the sample space is uniform, this definition is equivalent to the definition of average.
Example 5. Two cards are selected at random from a standard deck of cards. What is the expected
number of hearts?
A life insurance company charges customers a yearly premium of $140 for a $25,000 life insurance
policy. If the probability that a given customer will live through the next year is 0.998, what is the life
insurance company’s net earnings?
Expected values are often used in games to determine whether the game is “fair.” In a game situation,
we let X be the net winnings (profit) of the player. A game is considered “fair” when the expected net
winnings are 0, i.e., when E(X) = 0.
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Example 6. Visitors at a carnival pay $1.00 to play a game. The game consists of pulling a marble out
of a bag. If the marble is purple, the participant wins $5.00. If the marble is blue, the participant wins
$2.00. If the marble is green, the participant wins $1. However, if the marble is white, the participant
loses. Assume the bag contains 1 purple marble, 2 blue marbles, 3 green marbles, and 4 white marbles.
a) Draw a probability distribution for this game where X represents your net winnings.
b) Determine the expected winnings.
c) Is this game fair?
Example 7. A game consists of rolling a pair of fair 6-sided dice. The game costs 4 dollars to play.
If you roll the same number on both dice (a double), you win a dollars. Otherwise, you win nothing.
What value of a would make this game fair?
The expected value of the binomial distribution with n trials and probability of success p in a single
trial is np.
Example 8. Given that a binomial trial is repeated 300 times with a probability of success of 0.68, find
the expected value of this binomial trial.
Example 9. Assume that the expected number of successes in a binomial experiment consisting of 60
trials is 23. What is the probability of success?
3.3 Measure of Spread
Consider two date set {−60, −3, −2, −1, 1, 2, 3, 60} and {−2, −2, −2, −1, 1, 2, 2, 2}. They have the same
mean value and median. How distinguish them?
Let X denote the random variable that takes on the values x1 , x2 , · · · , xn , and let the associated probabilities be p1 , p2 , · · · pn . Then if µ = E(X), the variance of the random variable X, denoted by Var(X),
is
Var(X) = (x1 − µ)2 p1 + (x2 − µ)2 p2 + · · · + (xn − µ)2 pn .
The standard deviation, denoted by σ(X) is
σ(X) =
p
Var(X).
Finding Standard Deviation (and Variance) Using Your Calculator:
1. Press STAT, then EDIT.
2. In the first column, enter the values the random variable X takes on.
3. In the second column, enter the frequency or probability for each value of X.
4. Exit to the home screen.
5. Press STAT. Scroll to the right to CALC. Select 1-Var Stats (Option 1). Press L1 (2nd → 1). Press
the comma key. Press L2 (2nd → 2). Press ENTER.
6. The standard deviation is given by σx.
7. To find the variance, press VARS. Select Statistics... (Option 5). Select σx (Option 4). Press x2 .
Press ENTER.
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Example 10. Find the variance and standard deviation of the random variable X having the following
probability distribution.
−1
X
P (X)
0
1
2
3
1/6 1/3 1/4 1/12 1/6
Example 11. A certain puzzle manufacturer makes and sells puzzles. The data below shows how many
puzzles this company makes with di?erent numbers of pieces.
Number of Pieces 500
Number of Puzzles
400
1000 750 2000 300 1500 3000
200
350
150
275
300
75
Find the standard deviation and variance for the number of pieces in a puzzle.
The standard deviation and variance of the binomial distribution with n trials and probability of success
√
p in a single trial and q = 1 − p is npq and npq.
Summary:
E(X) = µ = x = mean = expected value
σ = standard deviation
Var(X) = σ 2 =variance
In a binomial experiment
E(X) = np
√
σ = npq
Var(X) = npq
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