Bernoulli trials - Columbia University
Bernoulli trials
An experiment, or trial, whose outcome can be classified as either a success or failure is performed.
X = 1 when the outcome is a success 0 when outcome is a failure
If p is the probability of a success then the pmf is, p(0) =P(X=0) =1-p p(1) =P(X=1) =p
A random variable is called a Bernoulli random variable if it has the above pmf for p between 0 and 1.
Expected value of Bernoulli r. v.:
E(X) = 0*(1-p) + 1*p = p
Variance of Bernoulli r. v.:
E(X 2) = 02*(1-p) + 12*p = p Var(X) = E(X 2) - (E(X)) 2 = p - p2 = p(1-p)
Ex. Flip a fair coin. Let X = number of heads. Then X is a Bernoulli random variable with p=1/2. E(X) = 1/2 Var(X) = 1/4
Binomial random variables
Consider that n independent Bernoulli trials are performed. Each of these trials has probability p of success and probability (1-p) of failure.
Let X = number of successes in the n trials.
p(0) = P(0 successes in n trials) = (1-p)n p(1) = P(1 success in n trials) = (n 1)p(1-p)n-1 p(2) = P(2 successes in n trials) = (n 2)p2(1-p)n-2
{FFFFFFF} {FSFFFFF} {FSFSFFF}
...... p(k) = P(k successes in n trials) = (n k)pk(1-p)n-k
A random variable is called a Binomial(n,p) random variable if it has the pmf, p(k) = P(k successes in n trials) = (n k)pk(1-p)n-k for k=0,1,2,....n.
Valid pmf: sum_{k=0}^n p(k) = 1 sum_{k=0}^n (n k)pk(1-p)n-k = (p+(1-p))n = 1
Ex. A Bernoulli(p) random variable is binomial(1,p)
Ex. Roll a dice 3 times. Find the pmf of the number of times we roll a 5.
X = number of times we roll a 5 (number of successes) X is binomial(3,1/6)
p(0) = (3 0)(1/6) 0(5/6)3 = 125/216 p(1) = (3 1)(1/6) 1(5/6)2 = 75/216 p(2) = (3 2)(1/6) 2(5/6)1 = 15/216 p(3) = (3 3)(1/6) 3(5/6)0 = 1/216
Ex. Screws produced by a certain company will be defective with probability .01 independently of each other. If the screws are sold in packages of 10, what is the probability that two or more screws are defective?
X = number of defective screws. X is binomial(10,0.01)
P(X 2) = 1 - P(X ................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- physics 121c mechanics
- stats review chapter 9
- computer organization and architecture cpu structure
- rethinking the sky
- p45871a gcse maths a p2 1ma0 2f jun15 edexcel
- bernoulli trials columbia university
- ca 527 tehama county coc application narrative
- off the rails how the met council misplans
- transport and inequality
- th bayifsmv bayrs 2126 g
Related searches
- columbia university graduate programs
- columbia university career fairs
- columbia university graduate tuition
- columbia university costs
- columbia university cost per year
- columbia university tuition and fees
- columbia university book cost
- columbia university cost of attendance
- columbia university graduate school tuition
- columbia university tuition 2019
- columbia university tuition 2020 2021
- columbia university neuroscience