Probability Distribution PDF

Probability Distribution: The list of all possible outcomes of a random variable along with their probabilities of occurrence is called probability distribution. Outcome(Marks) Probability 0 P(0) 1 P(1) 2 P(2) 3 P(3) 4...

Probability Distribution: The list of all possible outcomes of a random variable along with their probabilities of occurrence is called probability distribution. Outcome(Marks) Probability 0 P(0) 1 P(1) 2 P(2) 3 P(3) 4 P(4) 5 P(5) Outcome Probability H 0.5 T 0.5 Types of Random Variable: Discrete Random Variable: variable only assumes integer values. Ex: 1. No. of Copies of Book in a book shop (0, 1,2……) 2. A consumer can buy 0,1,2 Shirts Continuous Random Variable: variable assume both integer and non integer values over a range of values (interval). Ex: 1. Product Cost and Prices 2. Floor Area of office and shop 3. Amount of Rainfall Expected Value (Mean) and Variance of Random Variable: Discrete Random Variable: E(x) = ∑ xjP(xj ) Variance σ2 = ∑ (xj- E(x) )2 P(xj ) Continuous Random Variable:  E(x) =  xP(x) dx  Example Ex 2 Adoctor recommends a patient to take a particular diet for two weeks and there is equal chance for the patient to lose weight between 2 kgs and 4 kgs. What is the average amount the patient is expected to lose on this diet? Soln : P(x) = {1/2 when 2=4) =1-P(x= 0 or1 or2 or 3) Q.2 The number of misprints on a page of the Daily News Paper has Poisson distribution with mean 1.2. Find the probability that the no. of errors (a) on page four is 2; (b) on page three is less than 3 (e-1.2 = 0.3012) 1. 0.2168 2. 0. 8794  Normal Distribution: Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In graph form, normal distribution will appear as a bell curve. Characteristics of the Normal Distribution It is a continuous distribution It is a symmetrical distribution about its mean It is asymptotic to the horizontal axis It is unimodal Area under the curve is 1 The random variables following the normal distribution are those whose values can find any unknown value in a given range. For example, finding the height of the students in the school. These random variables are called Continuous Variables, and the Normal Distribution then provides here probability of the value lying in a particular range for a given experiment. Applications The normal distributions are closely associated with many things such as:  Marks scored on the test  Heights of different persons  Size of objects produced by the machine  Blood pressure and so on. For a normal distribution, 68% of the observations are within +/- one standard deviation of the mean (µ-σ to µ+σ), 95% are within +/- two standard deviations (µ-2σ to µ+2σ), and 99.7% are within +- three standard deviations (µ-3σ to µ+3σ). The general form of its probability density function is  1 ( x   ) 1 ( )2 f ( x)  e 2   2 Where П= constant 3.1416 e = constant 2.7183 µ= mean of normal distribution σ = standard deviation of normal distribution The normal distribution is described by its mean and standard deviation All normal distributions can be converted to a single distribution, the z distribution, using the formula: Z= x- µ/ σ A z score is the number of standard deviations that a value, x, is above or below the mean The z distribution is a normal distribution with a mean of 0 and a standard deviation of 1 Excel Formula: = NORM.DIST(x, mean, standard deviation, cumulative) Q.1 The lifetimes of certain kinds of electronic devices have a mean of 300 hours and standard deviation of 25 hours. Assuming that the distribution of these lifetimes, which are measured to the nearest hour, can be approximated closely with a normal curve (a) Find the probability that any one of these electronic devices will have a lifetime of more than 350 hours. (b) What percentage will have lifetimes of 300 hours or less? (c) What percentage will have lifetimes from 220 to 260 hours? Q.2 Most graduate schools of business require applicants for admission to take the Graduate Management Admission Council’s GMAT examination. Scores on the GMAT are roughly normally distributed with a mean of 527 and a standard deviation of 112. What is the probability of an individual scoring above 500 on the GMAT? Q.3 The weekly wages of 1000 workmen are normally distributed around a mean of Rs. 70 with a standard deviation of Rs. 5. Estimate the number of workers whose weekly wages are 1) Between Rs. 69 to Rs. 72 2) Less than Rs. 69 3) More than Rs. 72