400 Statistics Lecture 9 Section 2.4 Bernoulli Trials Bernoulli Experiment: A Bernoulli experiment is a random experiment, the outcome of which can be classified as success and failure. Let X be random variable of Bernoulli distribution. Outcome Probability Success 1 p Failure 0 q=1-p
Notation: X ~ Ber (p) pmf Mean E[X]=p
Variance Var[X]=pq
A sequence of Bernoulli Trials: A sequence of Bernoulli experiments are conducted independently and the probability of success (p) remains the same.
Ping Ma Lecture 9 Fall 2006
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Example: For a million of instant lottery tickets, suppose that 20% are winners, If 5 tickets are purchased, (1) what is the probability we have (0,1,0,1,0)?
(2) what is the probability we have (0,0,1,1,0)?
(3) What is the probability we have two winning tickets?
Binomial distribution 1. A Bernoulli experiment is performed n times 2. The trials are independent 3. The probability of success on each trial is a constant p; The probability of failure each on trial is a constant 1-p 4. The random variable X equals the number of successes in the n trials. X~Bin(n,p) ( b(n,p) ) n and p are parameters of binomial distribution.
Ping Ma Lecture 9 Fall 2006
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Example: What is the probability that we have at most two winning tickets among 5 tickets?
Cumulative distribution function (CDF) F(x)=P(X x)
Example: Your job is to examine light bulbs on an assembly line. You count the number of defective light bulbs after examining 10 light bulbs. Let X = number of defective light bulbs P (defective) = .2 N = 10 1. What is the probabil...