MA : Introductory Probability
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1 MA : Introductory Probability David Murrugarra Department of Mathematics, University of Kentucky Spring 2017 David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
2 Bernoulli Experiment A Bernoulli experiment is a random experiment, the outcome of which can be classified in one of two mutually exclusive and exhaustive ways say, success or failure. David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
3 Bernoulli Experiment A Bernoulli experiment is a random experiment, the outcome of which can be classified in one of two mutually exclusive and exhaustive ways say, success or failure. Let X be a random variable associated with a Bernoulli trial by defining it as follows: X(success) = 1 and X(failure) = 0. David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
4 Bernoulli Trials A sequence of Bernoulli trials occur when a Bernoulli experiment is performed several independent times so that the probability of success remains the same from trial to trial. David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
5 Bernoulli Trials A sequence of Bernoulli trials occur when a Bernoulli experiment is performed several independent times so that the probability of success remains the same from trial to trial. Let p denote the probability of success in each trial and q = 1 p the probability of failure. David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
6 Bernoulli Trials A sequence of Bernoulli trials occur when a Bernoulli experiment is performed several independent times so that the probability of success remains the same from trial to trial. Let p denote the probability of success in each trial and q = 1 p the probability of failure. Example Consider the experiment of flipping a fair coin five independent times. The probability of heads on any one toss is 1/2. David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
7 Binomial Probabilities We shall be interested in the probability that in n Bernoulli trials there are exactly j successes. David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
8 Binomial Probabilities We shall be interested in the probability that in n Bernoulli trials there are exactly j successes. Definition Given an n Bernoulli trial with probability p of success, the probability of exactly j successes is denoted by b(n, p, j). David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
9 Binomial Probabilities We shall be interested in the probability that in n Bernoulli trials there are exactly j successes. Definition Given an n Bernoulli trial with probability p of success, the probability of exactly j successes is denoted by b(n, p, j). Example Calculate b(3, p, 2). David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
10 Binomial Probabilities Let the random variable X equal the number of observed successes in n Bernoulli trials, then the possible values of X are 0, 1, 2,..., n. David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
11 Binomial Probabilities Let the random variable X equal the number of observed successes in n Bernoulli trials, then the possible values of X are 0, 1, 2,..., n. If x successes occur, where x = 0, 1, 2,..., n, then n x failures occur. David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
12 Binomial Probabilities Let the random variable X equal the number of observed successes in n Bernoulli trials, then the possible values of X are 0, 1, 2,..., n. If x successes occur, where x = 0, 1, 2,..., n, then n x failures occur. The number of ways of selecting x positions for the x successes in the n trial is ( ) n n! = x x!(n x)!. David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
13 Binomial Distribution f (x) = ( ) n p x (1 p) n x, x = 0, 1, 2,..., n. x These probabilities are called binomial probabilities, and the random variable X is said to have a binomial distribution. David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
14 Binomial Distribution f (x) = ( ) n p x (1 p) n x, x = 0, 1, 2,..., n. x These probabilities are called binomial probabilities, and the random variable X is said to have a binomial distribution. A binomial distribution will be denoted by the symbol b(n, p), and we that the distribution of X is b(n, p).the constants n and p are called the parameters of the binomial distribution. David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
15 Binomial Distribution f (x) = ( ) n p x (1 p) n x, x = 0, 1, 2,..., n. x These probabilities are called binomial probabilities, and the random variable X is said to have a binomial distribution. A binomial distribution will be denoted by the symbol b(n, p), and we that the distribution of X is b(n, p).the constants n and p are called the parameters of the binomial distribution. f (x) = 1 x S David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
16 Binomial Distribution Figure: Binomial density function. David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
17 Binomial Distribution f (x) = ( ) n p x (1 p) n x, x = 0, 1, 2,..., n. x David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
18 Binomial Distribution f (x) = ( ) n p x (1 p) n x, x = 0, 1, 2,..., n. x µ = E(X) = np. σ 2 = npq David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
19 Cumulative Distribution Function The cumulative distribution function or, more simply, the distribution function of the random variable X is F(x) = P(X x), < x <, For the binomial distribution the distribution function is defined by x ( ) n F(x) = P(X x) = p y (1 p) n y y y=0 where x is the floor or greatest integer less than or equal to x. David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
20 Binomial Distribution Figure: Binomial distribution cdf. David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
21 Binomial Distribution Example Consider the experiment of flipping a fair coin six independent times. What is the probability that exactly three heads turn up? David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
22 Binomial Distribution Example Consider the experiment of flipping a fair coin six independent times. What is the probability that exactly three heads turn up? b(6, 0.5, 3) = ( 6 3 ) ( 1 2 ) 3 ( ) 1 3 = David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
23 Binomial Distribution Example Consider the experiment of rolling a fair die four independent times. What is the probability that exactly one six? David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
24 Binomial Distribution Example Consider the experiment of rolling a fair die four independent times. What is the probability that exactly one six? We this experiment as a 4 Bernoulli trials with success: "rolling a 6" and failure: "rolling some number other than 6". David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
25 Binomial Distribution Example Consider the experiment of rolling a fair die four independent times. What is the probability that exactly one six? We this experiment as a 4 Bernoulli trials with success: "rolling a 6" and failure: "rolling some number other than 6". Then p = 1/6. David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
26 Binomial Distribution Example Consider the experiment of rolling a fair die four independent times. What is the probability that exactly one six? We this experiment as a 4 Bernoulli trials with success: "rolling a 6" and failure: "rolling some number other than 6". Then p = 1/6. b(4, 1/6, 1) = ( 4 1 ) ( 1 6 ) 1 ( ) 5 3 = David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring / 12
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