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Title: Distributions
Description: A statistical distribution is a mathematical function that describes the probabilities of all possible outcomes of a random variable. The type of distribution used to model a particular set of data depends on the nature of the data and the research question being addressed.
Description: A statistical distribution is a mathematical function that describes the probabilities of all possible outcomes of a random variable. The type of distribution used to model a particular set of data depends on the nature of the data and the research question being addressed.
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Discrete Probability
Distributions
Chap 5-1
Learning Objectives
In this chapter, you learn:
The properties of a probability distribution
To compute the expected value and variance of a
probability distribution
To calculate the covariance and understand its use
in finance
To compute probabilities from binomial,
hypergeometric, and Poisson distributions
How the binomial, hypergeometric, and Poisson
distributions can be used to solve business
problems
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc
...
g
...
Continuous variables produce outcomes that
come from a measurement (e
...
your annual
salary, or your weight)
...
Chap 5-3
Types Of Variables
Types Of
Variables
Ch
...
6
Chap 5-4
Discrete Random Variables
Can only assume a countable number of values
Examples:
Roll a die twice
Let X be the number of times 4 occurs
(then X could be 0, 1, or 2 times)
Toss a coin 5 times
...
Chap 5-5
Probability Distribution For A
Discrete Random Variable
A probability distribution for a discrete random
variable is a mutually exclusive listing of all
possible numerical outcomes for that variable and
a probability of occurrence associated with each
outcome
...
20
0
...
24
0
...
Chap 5-6
Example of a Discrete Random
Variable Probability Distribution
Experiment: Toss 2 Coins
...
25
1
2/4 = 0
...
25
Probability
4 possible outcomes
Let X = # heads
...
50
0
...
1
2
X
Chap 5-7
Discrete Variables
Expected Value (Measuring Center)
Expected Value (or mean) of a discrete
variable (Weighted Average)
N
µ = E(X) = ∑ X i P( X = X i )
i =1
Example: Toss 2 coins,
X = # of heads,
compute expected value of X:
X
P(X=Xi)
0
0
...
50
2
0
...
25) + (1)(0
...
25))
= 1
...
Chap 5-8
Discrete Random Variables
Measuring Dispersion
Variance of a discrete random variable
N
σ 2 = ∑ [X i − E(X)]2 P(X = X i )
i =1
Standard Deviation of a discrete random variable
σ = σ2 =
N
2
−
[X
E(X)]
P(X = X i )
∑ i
i =1
where:
E(X)
= Expected value of the discrete random variable X
Xi
= the ith outcome of X
P(X=Xi) = Probability of the ith occurrence of X
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc
...
25) + (1− 1)2 (0
...
25) = 0
...
707
Possible number of heads
= 0, 1, or 2
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc
...
A positive covariance indicates a positive
relationship
...
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc
...
g
...
Chap 5-22
Binomial Probability Distribution
(continued)
Observations are independent
The outcome of one observation does not affect the
outcome of the other
Two sampling methods deliver independence
Infinite population without replacement
Finite population with replacement
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc
...
Chap 5-24
The Binomial Distribution
Counting Techniques
Suppose the event of interest is obtaining heads on the
toss of a fair coin
...
In how many ways can you get two heads?
Possible ways: HHT, HTH, THH, so there are three
ways you can getting two heads
...
We need to be able to
count the number of ways for more complicated
situations
...
Chap 5-25
Counting Techniques
Rule of Combinations
The number of combinations of selecting X
objects out of n objects is
n!
n Cx =
X!(n − X)!
where:
n! =(n)(n - 1)(n - 2)
...
(2)(1)
0! = 1 (by definition)
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc
...
31!
31! 31 • 30 • 29 • 28!
= 31 • 5 • 29 = 4,495
=
=
31 C 3 =
3!(31 − 3)! 3!28!
3 • 2 • 1 • 28!
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc
...
, n)
n
= sample size (number of trials
or observations)
π = probability of “event of interest”
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc
...
5
1 - π = (1 - 0
...
5
X = 0, 1, 2, 3, 4
Chap 5-28
Example:
Calculating a Binomial Probability
What is the probability of one success in five
observations if the probability of an event of
interest is 0
...
1
n!
P(X = 1 | 5,0
...
1)1 (1 − 0
...
1)(0
...
32805
Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc
...
02
...
02
n!
π x (1 − π ) n − x
P(
Title: Distributions
Description: A statistical distribution is a mathematical function that describes the probabilities of all possible outcomes of a random variable. The type of distribution used to model a particular set of data depends on the nature of the data and the research question being addressed.
Description: A statistical distribution is a mathematical function that describes the probabilities of all possible outcomes of a random variable. The type of distribution used to model a particular set of data depends on the nature of the data and the research question being addressed.