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Page #1/10 A Presentation on DiFFERENCES B/W DiSTRiBUTiONS Brief introduction of Binomial, Hypergeometric, Poisson, Normal Distributions. Presented by NAIMA SABIR………..Roll # 909 ASMA SAJID……………Roll # 936 BCS-II Presented before Sir. YAHYA SAEED COMPUTER SCIENCE DEPARTMENT GCUF Page #2/10 Objective To provide easy understandability of all distributions so that one may easily judge where to apply which distribution. Differentiating b/w these distributions helps to apply the exact one. Page #3/10 OUTLINE Binomial Distribution. Normal/Gaussian Distribution. Poisson Distribution. Hypergeomtric Distribution. Definition Mathematically Why we use it Properties Daily life example Numerical Differences b/w all. Application of Hypergeometric Distribution(proceedings of annual Meeting of American Statistical Association Aug5-9,2001) Summary. Question & Answering Session. --------------------------- Page #4/10 Binomial Distribution HISTORY: The first one distribution, which was developed, was Binomial Distribution. Although there is some other statements of satiations, which tells that hypergeometric was the first one to be made but it, doesn’t seems to be true. Binomial experiment is that experiment in which out comes are always classified in success or in failure. Type:- This is distribution of Discrete type Definition: “A distribution based on Binomial experiment is called Binomial Distribution.” OR “The probability distribution of a numerical quantity whose value is determined by the implementation of Binomial experiment is called Binomial Distribution.” Mathematically: Its mathematical representation is given by; If “p” is probability of success & “q” is probability of failure then from n samples Binomial distribution is P (x; n, p, q) =nCx px qn-x Here x =0,1,2,3,4,5,. . . Why we use it?: When our possible outcomes are only two. when we have success or failure It describes the possible no. of times that a particular event will occur in a sequence of observations. The event is coded binary; it may or may not occur. Properties: 1. 2. 3. 4. 5. 6. The probability of success remains constant. Successive trails are independent. Experiment is repeated for a Large no. of times. Each trail outcome has two categories; success or fail. Its general form depends upon parameters p and q. It may also be applied to a finite population if it is not very small. Page #5/10 Features of binomial distribution:µ=np σ2=npq σ=√npq Mean Variance Standard Deviation Daily life implementation: A coin tossed during a cricket match. What is the probability that Pakistani captain gets head. When a researcher is interested in the occurrence of an event not in its magnitude. A baseball player’s batting average is 0.250.what is probability that he gets exactly one hit in his next 5 times at bat? Numerical: What is the probability of rolling two sixes & three nonsixes in 5 independent cast of a fair die? Solution Here, n=5 X=2 p+q =1 P=1/6 so q=1-(1/6)=5/6 Putting values in it P (2)=5C2 (1/6) 2 (5/6) 3 Numerical 2: A baseball player’s batting average is 0.250.what is probability that he gets exactly one hit in his next 5 times at bat? Solution Here p=0.25 q=1-p ,q=0.75 n=5 ,x=1 Using these values we get, P (1)=5C1 (0.25) 1(0.75) P=5 x 0.25(0.75) P=0.39550 Ans 5-1 Page #6/10 important Binomial approaches Poisson for large “n” and small” Binomial approaches Normal if no. of observations are large Hypergeometric Distribution Hypergeometric Experiment:An experiment in which a random sample is selected from a “known finite” population “without replacement”. Definition:“The probability distribution of hypergeometric experiment is called Hypergeometric Distribution.” OR “In general we are interested in the probability of selecting “x” successes from “k” items labeled “success”& “n-x” items labeled as failures. When a random sample of size “n” is selected from a finite population of size “N”.” Mathematically: It is denoted by h and ids given by, h(x; k, n,N)=kCx (N-k)C(n-x)/NCn Why we use it? When there are two events and we are interested in one of it. For example there are men and women in a particular firm and our interest is in men of that firm. Here we ‘ll use hypergeometric distribution. Properties: 1) 2) 3) 4) 5) Each trail is classified into success & failure in one of them we are concerned. Successive trails are dependant The probability of success will change after each trail. Experiment is repeated for a fixed no of time. If we set k/N=p, then the mean of hypergeometric distribution coincides with mean of binomial distribution. 6) When N is large hypergeometric approaches the “Binomial distribution”. Daily life implementation: For lotto games like “PowerBall” and “Gopher 5”binomial distribution doesn’t apply. Because in lotto games events are not independent. These are events without replacement. The selection of successive balls is not put back into the machine. Page #7/10 For “Power ball “ probability of first ball being a 1 is 1/53.if a 1 ball is selected, the probability of a 1 ball on second ball gets Zero. If 1 ball is not selected on first attempt, its selecting probability drops to 1/52 because there is one ball lesser in the machine. This is referred to as “sampling without replacement”. Numerical: What will be the probability of matching three of five white balls in “Powerball”game? Solution Here we have Total balls =N=53 No. of white balls =k=5 No. of balls that match a no. on our ticket =n=5 No. of balls drawn = x =3 So applying H.d H(3;5,5,53) =(5C3) ( 48C2 H(3;5,5,53) =10 x1128/2869685 =0.003931 Ans. ) / (53C5 ) Using Ms Excel A built-in function is used to calculate hypergeometric distribution called HYPEGEMDIST ………………………………. Page #8/10 Poisson Distribution Type:-This is a discrete type Distribution. Definition: “It is defined as the limiting case of Binomial distribution when “n” is very large & “p” is very small.” Mathematically: Its probability distribution is given by P(x=( e- µ µx ) /x! Why we use it? It is used when the probability of success is very very small. When the experiment in which we are concerned has very less success probability. Properties: 1) 2) 3) 4) 5) 6) 7) The probability of success is very small. Trails are independent In Poisson Mean & variance are equal MEAN=VARIANCE= µ Average is involved. Experiment is repeated for a fixed no. of times. For small values of meu, it is not symmetrical but skewed. It resembles the Binomial if the probability of an event is very small. Daily life implementation: It is widely used in biology. It used to find no. of bacteria on a plate It is used to calculate no. of miss prints in a 1000 or more page book. It was firstly defined for the description of no of deaths by hoarse kicking in Prussian army.(By Poisson in 1837). It is applied to the probability of mutations in a DNA Segment. It is also used to describe the distributions of counts detected from a Radioactive sample. Numerical: A secretary makes 2 errors per page on average. What is the probability that on next page she makes no errors ? Page #9/10 Solution Here so x=0. µ=2 P(0;2)=e-2 20/0! p(0;2)=0.13536 Ans NORMAL DISTRIBUTION Type:- it is a continuous type distribution History: - In 1733 Demoivre derived the mathematical equation of “Normal curve”. It is often referred to as “Gaussian Distribution” in honor of “Gauss (1777-1855). Definition: “It is a limiting case of Binomial Distribution when “n” is very large & “p” is moderate. Mathematically: if x is a normal random variable with mean µ & variance σ then equation of the normal curve is n(x; µ, σ) =e-1/2(x-µ/ σ )2/ σ√2∏ -∞< x <+∞ Why we use it? It is generally used when we are given σ or µ to find the area under the curve when we are given to find the specific area under the curve. It is always converted in to Standard Normal Distribution for find in area under the curve. Properties: 1) 2) 3) 4) 5) It is symmetrical if MEAN=MEDIAN =MODE If µ=0 and σ=1 we convert Normal in to Standard normal If n (a1<x<a2) then Ф (a2)-Ф (a1) If n (a>x) then 1- Ф (a) It is a distribution of infinite variables. Page #10/10 Differences B/w Distributions In this section we ‘ll discuss differences among these distributions Binomial Distribution Hypergeom etric Distribution Poisson Distributio n Normal Distribution “The probability distribution of hypergeometric experiment is called Hypergeometri c Distribution.” It is defined as the limiting case of Binomial distribution when “n” is very large & “p” is very small.” “It is a limiting case of Binomial Distribution when “n” is very large & “p” is moderate P(x=( e- µ µx ) /x! n(x; µ, σ) =e-1/2(x-µ/ σ )2/ σ√2∏ When we r given sigma or meu Definition “A distribution based on Binomial experiment is called Binomial Distribution.” Notation P (x; n, p, q) =nCx px qn-x h(x; k, n,N)=kCx k)C(n-x)/NCn When needed When possible outcomes are 2 Parameters Value of u Mean comparison Probability P,n Not exist Irrelevant We have two or more events and we r interested in one N, k, n Not exist Irrelevant When probability of success is very small µ Probability involves Probability involves With Replacement Without replacement Trails are Trails are independent independent Experiment is Experiment is repeated for a fixed repeated for a no. of time fixed no. of time Average involves Not concerned Not concerned Independent trial Experiment repeated for a fixed no. of time Not related Dependent variable exists Does not exist Replacement Trails No. of experiments Constant Probability of success remains constant Not remains constant (N- µ=np Mean =variance= µ µ, σ Any given Mean=median= mode Not concerned Not related