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Chapter 8. Probability Distributions: Part I
„ Chapter 7 will be covered as a section of
Chapter 8; This is the way I prefer to teach.
„ Suppose someone offer you the following
game (the Carnival Dice Game), will you be
willing to put down $1 to play the game?
„ You choose a number from {1,2,3,4,5,6}, then
throw a die three times; record the number of
times that your lucky number shows up.
„ If your number shows up once, the host pays
you $2; if the number shows up twice, you get
$3; if the number shows up in all three times,
you get $4. Otherwise, you lose your $1 bet.
The Colorado Lotto and the Powerball
„ To play
, you pay $1 and select 6 numbers from
1 to 42. If you match all 6 numbers you hit the
jackpot. There are three additional ways that you can
win. The prize of jackpot starts from $1m and varies.
„ To play
, you pay $1 and select 5 numbers
from 1 to 53 plus 1 number from 1 to 48. If you match
these 5+1 numbers you hit the jackpot. There are
eight additional ways that you can win. The prize of
jackpot starts from $10m and varies.
„ Should you play either lotto for a given jackpot prize?
If you have only $1 and want to win big, should you
play Colorado Lotto or the Powerball?
Decision-Making and Probability
Distributions
Random Variables
„ To make decision in a situation like those
„ If the outcome of an experiment can assume
mentioned above, one needs to know the
odds (probability) for each possible outcome.
„ A complete list of these probabilities is termed
a probability distribution in statistics.
„ With the probability distribution, a firm makes
the decision among choices (scenarios) using
the expected value. For an individual person,
the criterion is the expected utility since
people may be risk averse.
at least two possible values and there is no
certainty which value will be taken, then the
outcome of the experiment is called a random
variable. Examples. Non-random variables.
„ There are two types of random variables:
Discrete random variable: can assume only a
finite number of values.
„ Continuous random variable: can assume
values over an entire interval or a range.
„
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Probability Distribution of A
Discrete Random Variable
The Probability Distributions of
the Three Lotteries
„ We use a capital letter such as X for a random
variable and x for all the possible values that X can
assume.
„
„
Throw a die once, X = the number on the die, x = 1, 2, 3, 4,
5, 6.
Flip a coin 20 times, X = the number of heads, x = 0, 1, 2, …,
20.
„ The probability distribution is the sequence of prob.
P(X=x) for all x, or simply written as f(x).
„
„
„
Throw a die once, f(x) = 1/6 for x = 1, 2, …, 6.
We must have: 0 ≤ f(x) ≤1 and Σ f(x) = 1.
More examples: Ex. 7.2 through 7.8 on p.162-3.
Case
Case
Prize
Prob.
Match
None
$0
125/216
Match
One
$2
75/216
Match
Two
$3
Match
Three
$4
15/216
1/216
Case
Prize
Prob.:1
in
Match
6/6
Jack
5245786
Match
5/6
$556
24287
Match
4/6
$46
556
Match
3/6
$4
37
The probability of not wining anything?
97.1% for the Lotto and 97.2% for the
Powerball.
c
c
53 5 48 1
Prize
Prob.: 1 in
Match 5/5+1
J’pot
137744880
Match 5/5+0
$100k
2939678
Match 4/5+1
$5k
502195
Match 4/5+0
$100
12249
Match 3/5+1
$100
10685
Match 3/5+0
$7
261
Match 2/5+1
$7
697
Match 1/5+1
$4
124
Match 0/5+1
$3
80
c / 53 c5 48 c1
48 5
Mathematical Expectation and Variance
of a Probability Distribution
The Expectations of the Three Lotteries
„ You may have observed that the number of people
„ For the carnival dice game, its expectation or mean
buying Powerball tickets increases as the amount of
the jackpot gets larger. Why? This is because the
expected value rises.
„ For a random variable X, its mathematical
expectation is
„
µ ≡ EX = ∑ xf (x )
and its variance is
„
σ X2 = ∑ ( x − µ )2 f ( x )
Both EX and σ are the same as Chapters 3&4. For
f(x) = 1/6 with x ∈{1,…,6}, EX = 3.5 and σ = 1.71.
„
„
„
is EX = ($0)(125/216) + ($2)(75/216) + ($3)(15/216)
+ ($4)(1/216) = $0.921. The meaning?
For the Colorado Lotto with $1m jackpot, its mean is
EX = ($1m)(1/5245786) + … = $0.213. Variance?
For the Powerball with $10m jackpot, its mean is EX
= ($10m)(1/120526770) + … = $0.173. Variance?
All three games are called “unfair games” -- a fair
game is the game with EX = the bet you put down.
To make Lotto a fair game, the jackpot must be
$4.12m; the jackpot for the Powerball is $99.7m.
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The Binomial Probability Distribution
„ In business and economics, we often encounter
the following experiment: the so-called
Bernoulli trial in which
„
„
„
There is a fixed number of trials and all trial are independent;
There are only two possible outcomes: success and failure;
The probability of success (failure) remains the same
through all trial. Cardinal dice game vs. Lotto
„ Denote X the number of successes in n trials
and p the probability of success in each trial,
then the probability distribution is
P ( X = x ) ≡ f ( x )= n C x p x (1 − p )n − x
for x = 0,1, 2, …, n.
The Binomial Probability Distribution
„ If X assumes a binomial distribution, we
denote it by X ∼ B(n,p).
„ The use of Table V to find probabilities; the
histogram of B(n,p) (Figure 8.2).
„ The expectation, variance and standard
deviation of a binomial distribution:
µ ≡ EX = n ⋅ p
2
σ X = np(1 − p ) and σ X = np(1 − p )
„ Examples: Problem 8.46 and 8.48; Carnival
dice game; telemarketing (10 & 100 calls).
Other Probability Distributions
„ The binomial Distribution is the most important
discrete distribution, but there are others:
The hypergeometric distribution: the trials are
dependent; sampling with/without replacement;
„ The Poisson distribution: if the number of trials
is very large;
„ The multinomial distribution: there are more
than 2 possible outcomes in each trial.
„
„ HW: 7.3 and 7.4; 8.4, 9, 12, and 14. Also
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calculate EX, σ X and σ X of 8.5, 8.9 and 8.12.
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