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Unformatted text preview: Ex 2. Continuing with the circuit board example: Number of Defects (X) Frequency p(x) 50 0.5 1 30 0.3 2 10 0.1 3 10 0.1 a.Find E(X). b. Find σ X 2 . c. Find σ X . 1 Ex 3.Let X be the number of dots on the face turning up in rolling a sixsided fair die. a.) Describe the pmf. p(x) = b.) Find E(X). c) Find Var (X) . d) Find σ X . e) Write f(2) in terms of F(x). 2 Chapter 4 Section 1: The Bernoulli Distribution • Experiment consists of one trial with only two possible outcomes: Success/Failure • Probability of a success (p) We define a discrete random variable , X as follows: X = 1 if the observed outcome is a success = 0 if the observed outcome is a failure P(X=1)= p , P(X=0)= 1p The pmf of X is given by p(x)= P(X=x) = p x (1p) 1x We write X ~ Bernoulli (p). E(X)= Var(X)= Ex: Consider the experiment of rolling a die once and define X=1(success) if the die comes up 6. X~ 3 Section 2: The Binomial Distribution Suppose that we repeat a Bernoulli experiment n times in a way that the trials are independent, that is, the outcome of one particular trial does not influence the outcome of any other trials. If the success probability in the Bernoulli experiments is given by p and if X denotes the number of successes in n trials , then X is said to have the Binomial distribution with parameter. We write X~Bin(n,p). So, Binomial Experiment is a series of Bernoulli trials with the same success probability p. Characteristics of a Binomial Experiment 1. Experiment consists of n identical trials 2. There are only two possible outcomes on each trial: Success/Failure 3. Probability of a success (p) is the same for each trail 4. Trials are independent of each other. If these conditions are satisfied and X= the number of Successes in n trials, then X has a Binomial distribution with parameters n and p denoted X ~ Bin(n, p). 4 Ex. Determine if the following scenarios are examples of Binomial Experiments. A. A coin is flipped 10 times and the number of heads is observed. B. Ask 100 randomly selected students if they drank (or drink) when they were (are) underage. Make sure that the questioning is handled in a way that the respondent’s answer would be confidential. Count the number of students who answer yes to the question. 5 C. Select a SRS of 10 switches from a shipment of 10,000 of them. Count the number of switches in the sample that do not conform to specifications. Suppose that unknown to us, 1,000 of the switches are nonconforming. Rule of thumb for Independence Suppose each trial of an experiment can result in S or F and the sampling method is SRS, but the sampling is without replacement from a population of size N. If the sample size (the number of trials) n is at most 5% of the population size, the experiment can be analyzed as approximately a Binomial experiment (n, p)....
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This note was uploaded on 08/05/2011 for the course STA 3032 taught by Professor Kyung during the Summer '08 term at University of Florida.
 Summer '08
 Kyung
 Statistics

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