Probability Distributions

Binomial Distribution Definition: P(X=x)=(ncx)(p^x)(q^x), x=0,1,2,...n and p+q=1 where n is number of trails, p is probability of success, q is probability of failure x is number of success. Properties: 1. Introduced by J. Bernoulli. 2. Mean = np, variance = npq, 3. Mode = (n + 1)p 4. Mean > Variance 5. If p = ½, symmetric. If p < ½, B.D. is skewed to the right. If p > ½, skewed to the left. 6. Physical conditions.' i) trails are independent. ii) trails must be finite and small. iii) each trail contain two mutually exclusive events. 7. As p increases for fixed n, the B.D. shifts to the right. 8. As n decreases for fixed p, the B.D. shifts to the left. 9. Binomial coefficients are given by the Pascal’s triangle. 10. It is discrete distribution which gives the theoretical probabilities. 11. N.D. is limiting case of B.D. when n is large and p is small. 12. Additive property Problems: 1. An unbiased coin is tossed for 6 times. Find the probability of getting (i) 5 heads (ii) most 2 heads. 2. A die is tossed 10 times. If 2 or 3 is regarded as success, find the probabilities that i) 3rd time success ans: 0.2601 ii) at least 9 times success. 3. The probability of occurrence of occupational disease to a worker of a chemical factory is ¼. Find the probability that 2 out of 5 workers chosen at random will suffer from this disease. Ans: 135/512. 4. There are two defective pencils in a pack of dozen pencils. If 3 pencils are taken at random, find the probabilities that (i) at the most one pencil is defective ii) two pencils are defective. 5. The mean and variance of B.D. are 15 and 6 respectively. Find the values of n and p. 6. If on an average 8 ships out of 10 arrive safely to ports. Find the mean and standard deviation of ships returning safely out of a total of 400 ships. Ans Mean = 320, S.D. = 8 7. An unbiased coin is tossed 10 times. Find the Binomial distribution mean and variance. 8. Given mean = 60, S.D. = 6 find B.D. 9. Given mean and S.D of B.N.V. if n = 6 and 9 P(X = 4) = P(X = 2). Ans: mean = 1.5, S.D.= 1.061. 10. How many tosses of a coin are needed so that probability of getting at least one head is 0.875. Ans: P(X ≥ 1) = 0.875 →1-(1/2)n = 0.875. Therefore n = 3. 11. Variance of B.D. can’t exceed n/4. (or) Let x be a B.V. with parameter n and p. Find what value of p is var(x) is maximum if you assume that n is fixed? Poisson Distribution Def:P(X=x)=(e^-λ)(λ^x!)/x!, x = 0,1,2,... infinite and λ>0 Properties: 1. Poisson distribution is associated with the name of French Mathematician Simon Denis Poisson. 2. e = 2.7183, np = λ = finite (>0). 3. Mean = Variance = λ 4. P.D. is +ve skewed and Leptokurtoses. 5. As p → 0 then P.D. is J-shaped and uni modal 6. Mode:if λ integer then mode is λ,λ-1. if λ is not integer then mode is integer part of λ. 7. P.D. is limiting form of the B.D. when n is large, p is small and np = λ 8. P.D. is a discrete and it gives theoretical probabilities and theoretical frequencies. 9. Additive property Problems 1. The probability that a patient will get reaction of a particular injection is 0.001. 2000 patients are given that injection. Find the probabilities that (i) 3 patients will get reaction (ii) more than 2 patients will get reaction. Ans. 0.18, 0.135, 0.325 2. Between the hours of 2 and 4 P.M. the average number of phone calls per minute coming into the switch board of a company is 2.5. Find the probabilities that during one particular minute there will be i) no phone call at all ii) exactly 3 calls. Ans. 0.2135 3. There are 100 misprints in a book of 100 pages. If a page is selected at random, find the probabilities that i) there will be no misprint in the page ii) there will be 1 misprint Ans. 0.3679,0.3679 4. The probability that a blade manufactured by a factory is defective is 1/500. Blades are packed in packets of 10 blades. Find the expected number of packets containing (i) no defective ii) one defective iii) 2 defective blades. In a consignment of 10,000 packets 5. In P.D. 3 P(X=2) = P(X=4). Find mean ad variance. Ans 6.