Transcript Probability Distributions - HAAGA

Discrete Probability Distributions
1
Random Variable


Random experiment is an experiment with
random outcome.
Random variable is a variable related to a
random event
2
Discrete - Continuous


Random variable is discrete if it can take no
more than countable number of values
Random variable is continuous, if it can take
any value in an interval
3
Discrete Random Variables





The number of throws of a coin needed before a
head first appears
The number of dots when rolling a dice
The number of defective items in a sample of 20
items
The number of customers arriving at a check-out
counter in an hour
The number of people in favor of nuclear power in
a survey
4
Continuous Random Variables




The yearly income for a family
The amount of oil imported into Finland in a
particular month
The time that elapses between the installation of a
new component and its failure
The percentage of impurity in a batch of chemicals
5
Discrete probability distribution

Discrete random variable values and their
probabilities.
6
Fortune wheel
If the probability to win when
rolling a fortune wheel is
15% then the probability
distribution for the number
of wins in 5 rolls is:
number of wins
probability
0
44,3705%
1
39,1505
2
13,8178%
3
2,4384%
4
0,2152%
5
0,0076%
7
Two Dice
Probability
7/36
6/36
5/36
4/36
3/36
2/36
1/36
0
0
1
2
3
4
5
6
7
8
9
10 11 12 13
Sum of outcomes
8
Cumulative Distribution
Cumulative distribution function F(x) equals the
probability to get at most x.
x
2
3
4
5
6
7
8
9
10
11
12
F(x)
1/36
3/36
6/36
10/36
15/36
21/36
26/36
30/36
33/36
35/36
36/36
When playing two dice the sum of
outcomes lies between 2-12. Using
cumulative distribution we can easily find
probabilities for different events:
P(X<7) = 15/36  0,42
P(X>9) = 1 – 30/36 = 6/36  0,17
P(4<X<9) = 26/36 – 6/36 = 20/36  0,56
9
Expected Value

Expected value is just like the mean in empirical
distributions
Examples:
 When playing a dice the expected value equals 3,5
 Insurance company is interested in the expected value of
indemnities
 Investor is interested in the expected value of portfolio’s
revenue
10
Expected value calculation

The expected value for a discrete random variable
is obtained by multiplying each possible outcome
by its probability and then sum these products
11
Expected value example 1



Annual costs of an investment are estimated to be
100 000 per year for next 10 years.
Under boom estimated revenue is 180 000 per
year and under recession 110 000 per year.
Probability of boom is 0,40 and probability of
recession is 0,60.
Estimate the profitability of the investment.
12
Expected Value example 2


Assume a lottery with 1000 lottery and 31 winning
tickets. One ticket wins 500, ten tickets win 300 and
20 tickets win 100.
Define the ticket price so that the expected value of
the win is 55% of the ticket price.
13
Expected value example 3
According to manufacturer’s statistics the car model
needs repairs under warranty as follows:
No repairs for 50% of cars
On the average 100 euros repairs for 20% of cars
On the average 200 euros repairs for 25% of cars
On the average 500 euros repairs for the rest of the
cars
How much should the warranty increase the price of
the car?
14
Expected Value example 4


An arranger of a sports event wants to take a rain
insurance. The insurance price is defined using the
probabilities of rain and the amounts of possible
indemnities.
Define the price so that it is 40% higher than the expected
value of indemnity.
Rain (mm)
Probability
Indemnity
0-2
50%
-
3-5
30%
500
6-10
18%
2000
11-
2%
6000
15
Binomial Distribution Bin(n,p)
Binomial experiments satisfy the following:
The
experiment consists of a sequence of n
identical trials
All possible outcomes can be classified into two
categories, usually called success and failure
The probability of an success, p, is constant from
trial to trial
The outcome of any trial is independent of the
outcome of any other trial
16
Binomial Distribution Random Variables




The number of heads when tossing a coin for 50
times
The number of reds when spinning the roulette
wheel for 15 times
The number of defective items in a sample of 20
items from a large shipment
The number of people in favour of nuclear power in
a survey
17
Binomial distribution and Excel
You can use Excel to find probabilities related to
binomial distribution random variables (the number of
successes x in the n trials:
Probability
=BINOMDIST(x;n;p;0)
Cumulative probability =BINOMDIST(x;n;p;1)
18
Poisson distribution
Poisson experiments satisfy the following
 The probability of occurrence of an event is the
same for any two intervals of equal length
 The occurrence or non-occurrence of the event in
any interval is independent of the occurrence or
non-occurrence in any other interval
 The probability that two or more events will occur in
an interval approaches zero as the interval
becomes smaller
19
Poisson Distribution Random Variables





The number of failures in a large computer system
during a given day
The number of ships arriving at a loading facility
during a six-hour loading period
The number of delivery trucks to arrive at a central
warehouse in an hour
The number of dents, scratches, or other defects in
a large roll of sheet metal
The number of accidents at a crossroads during
one year
20
Poisson and Excel
You can use Excel to find probabilities related to
Poisson distribution random variables (the number of
occurrences x in an interval):
Probability =POISSON(x;;0)
Cumulative probability =POISSON(x; ;1)
 = the average number of occurrences in an interval
21
Continuous Probability Distributions
22
Normal Distribution
Many continuous variables are approximately
normally distributed
Measurement errors
Physical and mental properties of people
Properties of manufactured products
Daily revenues of investments
23
Normal Distribution
Normal distribution is defined by density function
area under density
function equals 1, area
represents probability
expected
value
24
Cumulative Probability Function

Cumulative function for x = area to the left of x =
probability to get at most x:
x
25
Standardized Distribution N(0,1)


Cumulative function values have been tabulated (in most
statistics textbooks) for normal distribution with expected
value 0 and standard deviation 1
This distribution is called standardized distribution and is
denoted N(0,1).
0
26
Standardized Distribution and Excel


Cumulative probability =NORMSDIST(z)
Random variable value z =NORMSINV(probability)
27
Standardizing
You can standardize any normal distribution N(,)
variable to a standardized distribution N(0,1) variable
SAME AREA! SAME PROB.!
x
z

z
0
x

28
Normal Distribution N(,) and Excel


Excel: =NORMDIST(x;;;1)
Excel: =NORMINV(cumulative probability;;)
29