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 3P(X) 0.10 0.30 0.20 0.20 0.15 0.05Find 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 processExample 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 successq= 1-p= probability of failurer= no of successes n= no of trialsFormula: P(X = r) = nCr p r (1-p)n-r nCr = ( n! / (n-r)! ) / r!

Binomial Probability DistributionExample-1:An insurance broker has five contracts and she believes that probability of making a sale for each contract is 0.40. Finda) 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= npWhere n= no of trialsP= 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

timeP(X)= Probability of x successes over a given timee= 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 axisWhenever average word is used in question, we use these equations:

μXbar= μσXbar= σ/ sqrt(n)

Normal Distribution Symbols• Let X= Value of random variableμ = Mean of distributionσ= Standard DeviationZ= 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?

ExampleA 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?