Download Probability Distribution

Statistical Inference
for
Managers
Lecture 3 Probability Distributions
Probability Distribution
Types of Probability Distributions:
(As discussed in first lecture)
1- Discrete Probability: Can have only a limited number of values.
2- Continuous Probability: Variable is allowed to take on any value within a
given range.
Random Variable: A variable which takes on different values as a result
of outcomes of a random experiment.
Expected Value:
Expected value is mean or average of the variable when probability
of the variable is also known.
Denoted by E(X)
E(X)= Σxi P(X)
E(X²)= Σxi² P(X)
Var(X)= E(X²)-(E(X))²
σx= Sqr(var(X))
Probability Distribution:
Example:
X
-2 -1 0 1
2
3
P(X) 0.10 0.30 0.20 0.20 0.15 0.05
Find E(X) and σ(X)?
Binomial Probability Distribution
Binomial Distribution: Describes a variety of processes of
interest to managers and represents discrete data that is the
result of an experiment called Bernoulli process
Example of Bernoulli process: Tossing a coin for a fixed
number of times and outcomes denoted by Binomial
Distribution.
• Each trial has two possible outcomes- H/ T or yes/ no or
success/ failure
• Probability of outcome is same for each trial
• Probability of outcome on one trial does not affect the
probability on other trials
Binomial Probability Distribution
Let
p= probability of success
q= 1-p= probability of failure
r= no of successes
n= no of trials
Formula:
P(X = r) = nCr p r (1-p)n-r
nCr = ( n! / (n-r)! ) / r!
Binomial Probability Distribution
Example-1:
An insurance broker has five contracts and she
believes that probability of making a sale for each
contract is 0.40. Find
a) The probability that she makes at most one
sale(0.337)
b) The probability that she makes between two
and four sales (inclusive) (0.653)
Binomial Probability Distribution
Example-2:
Suppose that a new computerized claims system has
been installed by a major health insurance company.
Only 40% of the claims require work by human claims
processor when this system is used. On a particular day,
100 claims arrived for processing. Assume that the
number of claims requiring work by a human follows a
Binomial Distribution. What is the probability that:
a) There are between 37 and 43(inclusive) claims that
require work by a human?
b) There are at most 38 claims that need the attention
of a human?
c) There are more than 42 claims that require work by
a human?
Binomial Probability Distribution
Mean and Standard Deviation of Binomial
Distribution:
Mean- μ= np
Where
n= no of trials
P= probability of success
Standard Deviation= sqrt(npq)
Poisson Distribution (Discrete Distribution)
Poisson Distribution:
Either n is very large or p is very small or both.
We are not given n and p separately, we are
given average of a process.
Probability function of Poisson Distribution can
be represented as:
P(x) = (e-μ) (μx) / xỊ
μx= λ= E(X)
Where
λ= mean or average no of occurrences (successes) over a
given time
P(X)= Probability of x successes over a given time
e= 2.71828
Poisson Distribution
Example:
Suppose that we are investigating the safety of a
dangerous intersection. Past police records indicate
a mean of five accidents per month at this
intersection. The no. of accidents is distributed
according to a poisson distribution and the Highway
Safety Division wants us to calculate the probability
in any month of exactly 0, 1, 2, 3 or 4 accidents.
Applications
 The no of failures in a large computer system
during a given day
 The no of delivery trucks to arrive at a central
warehouse in an hour
 The no of customers to arrive for flights during
each 15-minute time interval from 3p.m to 6p.m
on weekdays
Poisson approximation to Binomial
P(x)= (np)ͯ*e-ⁿᴾ
X!
Poisson approximation to Binomial
Example:
An analyst predicted that 3.5% of all small
corporations would file for bankruptcy in the
coming year. For a random sample of 100 small
corporations, estimate the probability that atleast
three will file for bankruptcy in the next year,
assuming that the analyst’s prediction is correct.
Normal Probability Distribution
Normal Distribution:
• Is applicable when the probability distribution is continuous.
• Due to its properties, it is applied to many situations in
which it is necessary to make inferences by taking samples.
Characteristics:
• Is Bell-shaped and shows the percentage of population
between 0 and z
• Mean, Median, Mode all lie at the centre
• Both tails never touch the axis
Whenever average word is used in question, we use these
equations:
μXbar= μ
σXbar= σ/ sqrt(n)
Normal Distribution Symbols
• Let
X= Value of random variable
μ = Mean of distribution
σ= Standard Deviation
Z= no of standard deviations from X to the mean
Z= X- μ
σ
Example
i. P(Z<1.31)= Φ(1.31)
ii. P(Z>2.30)= 1-Φ(2.30)
iii. P(Z<-2.13)= 1-Φ(2.13)
iv. P(-1.32<Z<1.58)= Φ(1.58)- (1-Φ(1.32))
Example: Average marks of students in a class are
60 with σ of 5. If a student is selected at random,
a) What is the probability that his marks are more than
70?
b) If 4 students are selected at random, what is the
probability that average marks are between 55 and
75? P(55<X<75)
Properties of Normal Distribution
Properties:
1. E(X+Y)= E(X)+E(Y)
2. E(aX+bY)= aE(X)+bE(Y)
3. Var(aX+bY)= a²var(X)+b²var(Y)
Example
A tin of juice contains some pieces of Fruit X and
some liquid juice of that fruit Y. If a tin contains on
average 30g of fruit pieces with standard deviation
of 10g and a tin contains liquid juice on average
900g with standard deviation of 50g.
a) Find probability that pieces of fruit in a tin are
between 25 to 35?
b) Find probability that sum of liquid and fruit
content is less than 1000?
Example
A company services copiers. A review of its record
shows that the time taken for a service call can be
represented by a normal random variable with mean
75minutes and standard deviation 20minutes.
a) What proportion of service calls take less than one
hour?
b) What proportion of service calls take more than 90
minutes?
c) The probability is 0.1 that a service call takes more
than how many minutes?