Tiêu đề XI - maths - chapter 8 - BINOMIAL DISTRIBUTION (134-146).pdf ✅

134 NARAYANAGROUP BINOMIAL DISTRIBUTION JEE-MAIN SR.MATHS-VOL-II W.E.1:- The value of 1 3;4, 4 B      is Sol: Here, r = 3, n = 4, p = 1/4, q = 3/4 p(x = 3) = 4 C3 3 1 1 3 4 4             = 3 64 W.E.2:- If the difference between the mean and variance of a binomial distribution for 5 trials is 5 9 , then the distribution is Sol: Here n = 5, np – npq = 5/9 np (1 – q) = 5/9  5 (p2 ) = 5/9; p=1/3, q = 2/3 then, binomial distribution is (2/3 + 1/3)5  The Characteristics of B.D. :  The mean of random binomial variate X is np. i.e., x np or np x   . , x x n p p n     The variance of r.b.v. x is npq. i.e., 2   npq Now 2 npq q x np          The S.D. of r.b.v. x is npq i.e.,   npq  In binomial distribution 2 2 x or x     i.e., np npq or npq np   W.E.3:-A symmetrical die is thrown four times and getting a multiple of 2 is considered to be a success. The mean and variance of success are Sol :Here, n = 4, p = 3/6 = 1/2, q = 1/2 Mean = np = 4(1/2) = 2 Variance = npq = 4 (1/2) (1/2) = 1  If 1 2 p q   , then the distribution is said to be a symmetrical binomial distribution. MODE : The mode is that value of variable with maximum probability.  The mode of B.D. depends on the value of np+p. CASE-1: If np+p=k, where k is an integer, then there will be two modes namely k & k-2. In this case the distribution is said to be a Bi-modal binomial distribution.  Bernoullian Trials (or) Bernoulli Trials Random trials which result either in the success or failure of an event A, with constant probability of success p and that of failure 1-p=q are called as bernoullian trials. For example: i) In tossing of an unbiased coin, if we consider getting head upwards as a success then the probability of success 1 2 p  . The probability of failure 1 1 1 2 2 q    and it is true for every trial. ii) In rolling of a symmetrical die, if we consider getting a face with 6 points upward as a success then 1 6 p  and 1 5 1 6 6 q    and it is also true for every trial.  Binomial Distribution : The probability of x successes in n independent bernoullian trials is given by   ; n x n x p X x c p q x    x n  0,1,2,3,..... ; p q   0, 0 and p q  1 and it is called as B.D. Here n & p are called as the parameters of B.D.  A discrete random variable x is said to follow B.D. with parameters n, p, if its probability mass function is given by   ; n x n x x p X x c p q    x n  0,1, 2,3,..... ; p q   0, 0 and p q  1. If X is the number of successes in n trials, then the probability distribution of X is No. of successes (x) P(X = x) n n 0 C q P 0 n n 1 1 C q P 1  n n r r C q P r  n 0 n C q P n 0 1 r n  The above probabilities are various terms of the binomial expansion   0 1 1 0 1 n n n n n q p c q p c q p     2 2 2 ..... ....... ...... n n n n r r n n n n r n c q p c q p c q p           The originator of B.D. was James Bernoulli (1654- 1705) and so it is also some times called as Bernoulli distribution.  B r n p  ; ,  means p X r    BINOMIAL DISTRIBUTION SYNOPSIS
NARAYANAGROUP 135 JEE-MAIN SR.MATHS-VOL-II BINOMIAL DISTRIBUTION CASE-2: If np+p=k+f, where k is an integer and f is a proper fraction then there will be only one mode namely k. i.e., the integral part of np+p will be the mode. In this case the distribution is said to be uni-modal binomial distribution. W.E.4:- In a binomial distribution with n=10, 2 5 p  , the mode of the B.D. is Sol: Here np + p = 10 (2/5) + (2/5) = 22/5 = 4 + (2/5); Mode = 4  If we consider n independent bernoullian trials as one experiment and if we repeat such an experiment N times, then the expected frequency or the theoretical frequency of x successes is given by   .   . , 0,1,2,3,.... n x n x x f X x N p X x N c p q x n       and this is called as Binomial frequency distribution. W.E.5:- Out of 10,000 families with 4 children each, find the expected number of families all of whose children are daughters Sol: Here, p = q = 1/2, n = 4, N = 10,000 Req. = N × P (x = 4) =(10,000) 4 C4 . 4 1 2       = 625  In a binomial distribution if ' '  is Mean and ' '  is S.D. then 0     (OR) [0,  ) W.E.6:- In a binomial distribution AM=3, variance=4. The intervel of standard deviation is Sol :Here, Mean (  ) = 3   [0, 3)  In binomial distribution P(X=r) is maximum when r np    (where [ ] is greatest integer function) W.E.7:- Suppose X follows binomial distribution with parameters n = 100 and 1 3 p  then P(x=r) is maximum when r is Sol: p x r    is maximum at r = [np] (where [ ] is greatest integer function)  r = 1 100 3        = [33.33] = 33 W.E.8:- In a binomial distribution 1 , 4 B n p       , if the probability of at least one success is greater than or equal to 9 10 , then n is greater than (AIEEE-2009) Sol: We have 9 ( 1) 10 P x    9 1 ( 0) 10    P x  0 0 1 3 9 1 4 4 10 n n C               9 3 1 10 4 n           3 1 4 10 n              Taking log to the base 3/4, on both sides, we get 3/ 4 3/ 4 3 1 log log 4 10 n              10 3/ 4 10 log 10 log 10 3 log 4 n           = 10 10 1 log 3 log 4    10 10 1 log 4 log 3 n   1. The number of parameters of B.D. are 1) 4 2) 3 3) 2 4) 1 2. If X B  (20, 1 / 2). The variance is 1) 3 2) 4 3) 5 4) 6 3. If for a Binomial distribution, μ = 10 and 2  = 5, then P(X > 6) is 1) 7 2 0 2 0 1 2 0 2 r r C   2) 2 0 2 0 6 1 2 0 2 Cr r   3) 2 0 1 9 7 1 2 0 2 C r r   4) 2 0 1 9 6 1 2 0 2 C r r   4. Let X be a B(2, p) and Y be an independent B(4, p). If P(X  1) = 5/9, then P(Y  1) is 1) 61 65 2) 65 81 3) 1 2 4) 1 3 C.U.Q.