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Binomial

# Binomial - To Fnd the probability P Y ≤ y use the sum...

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Binomial Distribution A discrete random variable is said to have a binomial distribution if The experiment involves a fixed number of independent and identical trials, n . Each of the n trials has two possible outcomes, success and failure. The probability of success on each of the n trials is the same, p . Therefore, the probability of failure on each of the n trials is 1 - p = q . The random variable Y is defined as the total number of successes in the n trials. The parameters for a binomial random variable Y are the number of trials n and the proba- bility of success on each trial p . The probability distribution function of the binomial random variable Y is p ( y ) = parenleftBigg n y parenrightBigg p y (1 - p ) n - y for y = 0 , 1 , . . . , n The theoretical mean of the binomial random variable Y is μ = E ( Y ) = np The variance of the binomial random variable Y is σ 2 = V ( Y ) = np (1 - p ) = npq Working with binomial random variables in R. Finding probabilities. To find a probability P ( Y = y ) = p ( y ) for a single value y , the command in R is

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Unformatted text preview: To Fnd the probability P ( Y ≤ y ), use the sum command to add up all p ( y ) values for y between and including 0 and y . sum(dbinom(0:y,n,p)) To Fnd the probability P ( y 1 ≤ Y ≤ y 2 ), use the sum command to add up all p ( y ) values for y between and including y 1 and y 2 . sum(dbinom(y1:y2,n,p)) To Fnd the probability P ( Y ≥ y ), use the sum command to add up all p ( y ) values for y between and including y and n 1 sum(dbinom(y:n,n,p)) Finally, to get a list of all p ( y ) values for a particular binomial random variable Y , use the dbinom command to list all p ( y ) for values of y between and including 0 and n . dbinom(0:n,n,p) • Generating observed values. You can also use R to generate observed values from a binomial distribution with n and p . The command is rbinom(numobs, n,p) where numobs is the number of observed values you would like to generate. 2...
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Binomial - To Fnd the probability P Y ≤ y use the sum...

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