MA305 Mean, Variance Binomial Distribution By: Prof. Nutan Patel Asst. Professor in Mathematics IT-NU A-203 patelnutan.wordpress.com MA305 Mathematics for ICE1

Probability Function If for random variable X, the real valued function f(x) is such that P(X=x) = f(x) Then f(x) is called probability function.  If X is a discrete random variable then its probability function f(x) is discrete probability function. It is called probability mass function.  If X is a continuous random variable then its probability function f(x) is continuous probability function. It is called probability density function. MA305 Mathematics for ICE2

Expected value MA305 Mathematics for ICE3

MEAN MA305 Mathematics for ICE4

Variance Variance characterizes the variability in the distributions, since two distributions with same mean can still have different dispersion of data about their means. Variance of R.V. X is Remark: MA305 Mathematics for ICE5

Standard Deviation (S.D.) S.D. Denoted by , is the positive square root of variance. Ex: Prove that (a) E(cX)=c E(X) (b) E(X+c) =E(X)+c Ex: Prove that (a) Var(cX) = c 2 Var(X) (b) Var(X+c)= Var(X) MA305 Mathematics for ICE6

Moment Generating Function (mgf) MA305 Mathematics for ICE7 Where is the usual moment of order r about the origin, is the coefficient of

M.g.f. of the distribution about any other value x=a is defined as Moments about mean is known as central moments. MA305 Mathematics for ICE8

Bernoulli Trail Experiments Suppose that you toss a coin ten times. What is the probability that heads appears seven out of ten times? A student guesses at all the answer on a ten MCQs quiz. Such problems involved repeated trials of an experiment with only two possible outcomes: heads or tails, right or wrong, win or loss, so on.. We classify the two outcomes as success or failure. When outcomes of an experiment are divided into two parts, it is called the situation of dichotomy. MA305 Mathematics for ICE9

Properties of Bernoulli Experiment 1.The experiment is repeated a fixed number of times (n times). 2.Each trial has only two possible outcomes: success and failure. The outcomes are exactly the same for each trial. 3.The probability of success remains the same for each trial. (probability of success is p and probability of failure q=1-p). 4.The trials are independent. 5.We are interested in the total number of successes, not the order in which they occur. MA305 Mathematics for ICE10

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Binomial Distribution MA305 Mathematics for ICE14

Application of Binomial Distribution 1. In Quality control charts (fraction defective or number of defective per sample) 2. Useful for insurance companies. 3. It is very useful in the application pertaining to behavioural sciences. 4. In research field where dichotomy is there, this distribution is used. 5. Estimation of reliability of systems. MA305 Mathematics for ICE15

Ex: Find the binomial distribution for n=4 and p= MA305 Mathematics for ICE16 XP(X=x)P(X)

Mean, Variance of Binomial Distribution MA305 Mathematics for ICE17

EX: Form the binomial distribution of the experiment of tossing a coin six times and counting the number of heads. EX: Compute mean, variance and s.d. of followings 1.n=50, p=0.4 2.N=600, p= N=470, p=0.08 MA305 Mathematics for ICE18

EX: If 10% of the rivets produced by a machine are defective, find the probability that out of 5 rivets chosen at random (i) none will be defective, (ii) one will be defective, and (iii) at least two will be difective. Ans: , , MA305 Mathematics for ICE19