# Probability Distribution

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Probability Distribution
Pooja Kansra LHSB

Random Variable A random variable(rv) is a numerical description of the outcome of an experiment. A rv is a variable which gives you a specific value for every outcome of an experiment Example: Consider an experiment to measure the number of customers who arrive in a shop during a timer interval of 2 minutes. The possible outcome may vary from 0 customers to n customers. These outcomes are the values of the random variable. The outcome vary from trial to trial. Probability Distribution

Types of random variable
Discrete r.v.: can take on some values within the range or r.v. which assumes either a finite number of values or a countable infinite number of possible values. Consider e.g. given in previous slide. Continuous r.v.: can take any value within a range or r.v. that assumes any numerical value in an interval or can take values at every point in a given interval. For eg. Time taken for services at a service station, time taken can be 2, 3, 2.34 minutes, and so on. Discrete (Continuous) r.v.: the number of outcome (not) are countable or (in) finite. Probability Distribution

Example of Random Variable
What are the outcomes to the following experiments? Are they continuous or discrete random variables? Experiment: Take a 20 question multi-choice examination! Experiment: Fill a bottle of olive oil. Experiment: Phone calls to an insurance claims department. 1 call Time between calls Both depending on how you frame the questions!! Discrete Continuous Probability Distribution Discrete Continuous

Classification of Distribution
Theoretical or Probability distribution Discrete Continuous Observed distribution Probability Distribution

Probability Distributions
A probability distribution is a listing of all possible outcomes (values) of an experiment together with the chance of occurring. A probability distribution gives the probability for each outcome (value) of a random variable. [Strictly speaking, these definitions apply more to discrete probability distributions as continuous ones can take on many values] Discrete probability distribution have a finite number of values. Important example: Binomial and Poisson Continuous distributions have a infinite number of values. Important example: Normal and Exponential Probability Distribution

Binomial Distribution
….. arises in many real world situations. It is appropriate where there are n identical trials, and we want to know the probability of getting x successes. The following conditions (assumptions) must hold. The n trials should be identical. The trials should be independent. Two possible outcomes at each trial: “success” & “failure”. The probability of success (p) is constant for each trial. Having recognized the problem, how do we apply it? Probability Distribution

PDF P(r) = ncr pr qn-r Probability Distribution

A simple Binomial Experiment
A coin is flipped 4 times. Let a head be defined as a success. The number of successes is a random variable, range 0 to 4. There are 16 possible outcomes (no. of heads in brackets): Possible results X (Prob. of getting head) P(X) TTTT 1/16 HTTT, THTT, TTHT, TTTH 1 4/16 HHTT, HTHT, HTTH, THHT, THTH, TTHH 2 6/16 HHHT, HHTH, HTHH, THHH 3 HHHH 4 Probability Distribution See Sheet No. 1 of Excel File: Probability Distribution – Vishal Sarin

Why Histogram? Probability Distribution

Applying the Binomial Formula
Probability Distribution

The Travelling Salesman
Q: A sales rep has observed that 1 in 5 potential customers he visits makes a purchase. He visits on average 5 customers per working day. What is the probability that he makes: Exactly 1 sale? Fewer than 2 sales? What is the probability of at least 3 sales? Solution to (i) Probability Distribution

The Cumulative Binomial Distribution
Sometimes we don’t want the probability of getting exactly ‘x’ successes, but the chance of getting ‘x’ successes or fewer (or x successes or more). In this case, we simply add up all component probabilities. Solution to (ii): P(x<2) = P(x=0) + P(x=1) Probability Distribution

Salesman cont….. Probability Distribution

Properties of BD Probability Distribution

The Poisson Distribution
Useful for estimating the number of occurrences of an event (x) within a specific interval (e.g. time, distance, area) Occurrence of goal in 90 minutes of football. And approximating the Binomial when ‘n‘ is large and ‘p’ is small. Developed by Simeon Poission….. but perfected by McDonalds and AT&T! Mean and Variance of PD are always equal i.e. =mean=variance=np PD is used when we are dealing with the problems where probability of success is very small i.e. p0 and q1 and n  Probability Distribution

The Poisson Distribution
Which of the following problems are best described as a Poission Probability distribution, and which by a Binomial distribution? The number of…. Heads when flipping five coins? Times lightning flashes in a day in July? Repairs on a given stretch of motorway? Vegetarian meals required on a 20 seater airplane? Hint: is there a natural upper limit? Probability Distribution

P.D. Two properties/assumptions must be satisfied:
Events occurs an average of  times during the interval. The event occurs at the same average rate throughout the interval. E.g.: the average number of goals in football matches is 2.12 (thus the average number of goals in half a football match is 1.06). The number of events that occur in an interval is independent of the number of events in any other non-overlapping interval. E.g.: the number of calls received at an office in the first hour does not influence the number of calls received in the second hour. Probability Distribution

If  is the average number of times the event is expected to occur in the interval, then the probability of the event actually occurring X times in the interval is : P(X=x) =  𝒙 𝒆 − 𝒙!  is the poisson distribution parameter. Where e is a constant, such that e= If e is raised to the power 2, e2 = But e is raised to a negative power e - = 1/ e - So if e is raised to the power -2, e -2 =1/ = Probability Distribution

Constants of PD Mean= m Variance= m; mean=variance
ß1= 1/m (measure of Skewnes) ß2= 3+1/m ( Measure of kurtosis) Probability Distribution

See pre determined table in books.
Example: If the average number of goals scored during a 90 minute football match is 2.1, what is the probability exactly 5 goals will be scored during a game? P(X=x) =  𝒙 𝒆 − 𝒙! = 𝟓 𝒆 −2.1 𝟓! = The number of goals is the random variable. We are interested in the chance of exactly 5 goals being scored when on average 2.1 goals are scored. Ans: 4 in 100 Mind you all the time computer is not available see “e” table for fraction values. See pre determined table in books. Probability Distribution

PROBLEMSSSSSSS The mean of poisson distribution is Find the other constants of PD. Ten percent of the tools produced in a certain manufacturing process turn out to be defective. Find the probability that in a sample of 10 tools chosen at random, exactly two will be defective.e-1= 0.184 Suppose on an average 1 house in 1000 in a certain district has a fire during a year. If there are 2000 houses in that district, what is the probability that exactly 5 houses will have a fire during the year? ,e=2.7183 Probability Distribution

Waiting in Line….. The number of people arriving at a bank is monitored for each 5 minute interval in the morning on a given day, and the mean was found to be 2.8 people. The same exercise was undertaken in the afternoon and found to be 2.1. What is the probability that: One person will arrive in a 5 minute interval in the morning, and similarly the afternoon? More than one person will arrive in a 5 minute period in the morning and similarly the afternoon. Probability Distribution

Waiting in Line….. P(X=x) = 2.8 𝒙 𝒆 −2.8 𝒙! = 0.1702
Poisson Probability Function Morning (AM) Evening (PM) P(X=x) = 𝒙 𝒆 −2.8 𝒙! = P(X=x) = 𝒙 𝒆 −2.1 𝒙! Probability Distribution

Problem of another type!
A manufacturing firm has 30 machines. The probability that any one of them will not function during a day is What is the probability that exactly two machine will be out of order on the same day? Solution: ? In this type of problem value of  is not provided directly. But we have to remember that  = np, rest of the sum be solved as usual. Probability Distribution

Summary Discrete Probability Distribution
The Binomial is used for calculating the probability of x successes in n independent trials. At each trial, there should be only two possible outcomes. The chance of success is given by p, and chance of failure is given by (1-p). The poisson distribution is used for estimating the number of occurrences of an event in a specific interval (e.g. time) The only parameter needed is  which indicates the average number of occurrence of the event in a given interval. See Excel sheet for similarity between PD and BD. B.D. has two parameters i.e. , , where PD has one parameter i.e.  . Statisticians most often use n larger than or equal to 20 and p less than or equal to 0.05 as the right case of approximating binomial distribution by the Poisson distribution. Probability Distribution

When it is not in our power to determine what is true, we ought to follow what is most probable.
-Descrtes Probability Distribution

Normal Distribution Sometimes called the Gaussian Distribution or Normal Curve It describes many real life situations….. Height, weight, intelligence of people Lifetime of machine parts, car tyres and production output. The concept of SIX SIGMA is also based on the properties of Normal curve. There are many normal distributions each with a different mean () and a different standard deviations (). Notation: N(, ), e.g. N(10,5) Probability Distribution

Properties of Normal Distribution
Is a continuous probability distribution (density function) On the horizontal axis any value in that interval of X can occur. In contrast, the Binomial is a discrete distribution with only integer value. The mean () determines the position of distribution. The larger the standard deviations (), wider/flatter the curve. With information of the  and , we have everything we need to know about a specific normally distributed variable. f(x)= 1  2∏ 𝑒 −1/2 𝑥−  2 Probability Distribution

Some important characteristics
Bell shaped normal curve has a single peak; thus, this is unimodal. Mean=Median= Mode, which lies at the centre of the normal curve. Mirror image. Two tails extend indefinitely, but never touch the horizontal axis. The SD is scatteredness of the normal curve. Sum of probability under normal curve is always 1. Probability Distribution

Area Property Approx. 68% of the values of a random variable lie within ± 𝜎 from the mean. Approx. 95.5% of the values of a random variable lie within ±2𝜎 from the mean. Approx. 99.7% of the values of a random variable lie within ±3𝜎 from the mean. Probability Distribution

Standard Normal Prob. Distribution (Z)
Every unique pair of Mena and SD describes a different normal distribution, this makes the analysis more difficult as this requires many normal tables. It is neither possible nor necessary. Then HOW? By using mechanism of Z, which convert all the normal distributions into a single distribution – z distribution. Z= 𝒙−𝝁 𝝈 , (this distribution has 0 mean and 1 S.D.) Z score can de defined as the number of standard deviation that a value, x is above or below the mean of the distribution. From z formula, this is clear that if the value of x is less than the mean, the z score is negative, otherwise positive. If the value of x is equal to the mean, the z score is zero. We require Z table to see the Z-score. Probability Distribution

Probability Distribution

Who will solve this question?
Let’s Do! The average earning per share (EPS) of a group of companies has a normal distribution with mean ` 25 and standard deviation ` 5. What is the probability that: A company selected at random from this group will have EPS in excess of ` 35? ……. between ` 20 and ` 40? ……… less than ` 20? …………….. less ` 30? Probability Distribution Who will solve this question?

OK! This time I will ……………..!
Probability Distribution

Roubel Tyres Roubel tyres company wishes to set a minimum mileage guarantee on its new MX200 tyre. Tests reveal the mean mileage is 67,900 with the standard deviation of 2,050 miles and that the distribution of miles follows the normal probability distribution. Roubel wants to set the minimum guaranteed mileage so that no more than 4 percent of the tires will have to be replaced. What minimum guaranteed mileage should Roubel announces? Ans: 64,312 miles Probability Distribution

One must work and dare if one really wants to live.
Probability Distribution