Probability Distributions

1
Probability Distributions

• Random Variable
• A numerical outcome of a random experiment
• Can be discrete or continuous
• Generically, x
• Probability Distribution
• The pattern of probabilities associated with all
  of the random variables for a specific experiment
• Can be a table, formula, or graph
• Generically, f(x)
• Examples
• Binomial (but wont cover here)
• Uniform
• Normal or bell-shaped distribution

2
Birth of a Distribution
Class Width 10
Cyberland Wages
3
Birth of a Distribution
Class Width 5
4
Birth of a Distribution
Class Width 2
5
Birth of a Distribution
Class Width 1
6
Birth of a Distribution
Class Width Very Small
7
Uniform Distribution
f(x)
Area 1
1 / (b-a)
x
a
b
8
Normal Distribution
Bell-shaped, symmetrical distribution
f(x)
x
9
Normal Distributions
?1
?2
?3
? 5
12
-2
10
Normal Distributions
Same ?,Different ?
11
Normal Distributions
68.26
?
??
?-?
12
Normal Distributions
95.44
?
?2?
?-2?
13
Normal Distributions
99.72
?
?3?
?-3?
14
Standard Normal Distribution
?z 0
?z 1
If x has a normal distribution
z
0
15
t Distribution
Specific thickness depends on degrees of freedom
Looks like a normal distribution,
but has thicker tails
3.5
-3.5
0
16
t Distribution
5 d.f. 10 d.f. 30 d.f. 100 d.f. ? d.f (normal)
Specific thickness depends on degrees of freedom
3.5
-3.5
0
17
Find the Probabilities

• P(z gt 2.36)
• P(t gt -1.02) with 5 degrees of freedom
• P(-0.95 lt z lt 1.93)
• P(-0.95 lt t lt -0.07) with 100 degrees of freedom
• Find z such that P(z lt z) 0.719
• Find z0.025 such that P(z gt z0.025) 0.025
• Find t0.025 such that P(t gt t0.025) 0.025 with
  5 degrees of freedom

18
z?/2
Standard Normal Distribution (z)
P(z lt -z?/2)) ?/2
P(z gt z?/2) ?/2
P(-z?/2 lt z lt z?/2) 1 - ?/2
0
-z?/2
z?/2
19
z?/2 for ?0.05
Standard Normal Distribution (z)
P(z lt ) 0.025
P(z gt ) 0.025
P( lt z lt ) 0.95
0
-z0.025
z0.025
?
?
20
t?/2 for ?0.05, df5
t distribution with 5 degrees of freedom
P(t lt ) 0.025
P(t gt ) 0.025
P( lt t lt ) 0.95
0
-t0.025
t0.025
?
?
21
?2 Distribution
Specific skewness depends on degrees of freedom
0
22
?2 Distribution
Specific skewness depends on degrees of freedom

• 5 d.f
• 10 d.f
• 15 d.f

0
23
?2 Distribution
10 d.f
P(?2 gt 18.307) 0.05
P(?2 lt 18.307) 0.95
0
18.307
24
F Distribution
Specific skewness depends on a pair of degrees of
freedom (df1, df2)
0
25
F Distribution
P(F lt 3.02) 0.95
9 and 10 d.f
P(F gt 3.02) 0.05
0
3.02
26
Probability Distributions
Normal t
?2
Different shapes and dfs, but SAME LOGIC !
F
27
In Excel

• To find probability above a value x
• 1-NORMSDIST(x)
• TDIST(x,df,1) 11-tail
• CHIDIST(x,df)
• FDIST(x,df1,df2)
• To find value with p above (e.g., 0.05)
• NORMSINV(p)
• TINV(p,df)
• CHIINV(p,df)
• FINV(p,df1,df2)

28
Word Problem

• From past experience, the management of a
  well-known fast food restaurant estimates that
  the number of weekly customers at a particular
  location is normally distributed, with a mean of
  5000 and a standard deviation of 800 customers.
• What is the probability that on a given week the
  number of customers will be between 4760 and
  5800?
• What is the probability of a week with more than
  6500 customers?
• For 90 of the weeks, the number of customers
  should exceed what amount?