Thursday, July 30, 2009

First year statistics + prob for eng?

In order to verify the accuracy of their financial accounts, companies use auditors on a regular basis to check the accuracy of the accounts entries. The companies accountants make errors in 5% of the entries. If an auditor randomly checks 3 entries, and the random variable Y is the number of errors detected by the auditor, then





a) Find the probability distribution of the random variable Y,





b) Find E(Y) and Var(Y),





c) And the probability that more than one error will be found by the auditor?





Step by step please.

First year statistics + prob for eng?
Let X be the number of entries with errors. X has the binomial distribution with n = 3 trials and success probability p = 0.05





In general, if X has the binomial distribution with n trials and a success probability of p then


P[X = x] = n!/(x!(n-x)!) * p^x * (1-p)^(n-x)


for values of x = 0, 1, 2, ..., n


P[X = x] = 0 for any other value of x.





The probability mass function is derived by looking at the number of combination of x objects chosen from n objects and then a total of x success and n - x failures.


Or, in other words, the binomial is the sum of n independent and identically distributed Bernoulli trials.





X ~ Binomial( n , p )





the mean of the binomial distribution is n * p = 0.15


the variance of the binomial distribution is n * p * (1 - p) = 0.1425


the standard deviation is the square root of the variance = √ ( n * p * (1 - p)) = 0.3774917





The Probability Mass Function, PMF,


f(X) = P(X = x) is:





P( X = 0 ) = 0.857375


P( X = 1 ) = 0.135375


P( X = 2 ) = 0.007125


P( X = 3 ) = 0.000125





P(X ≥ 1) = 1 - P(X %26lt; 1)


= 1 - P(X = 0)


= 1 - 0.857375


= 0.142625





P(X %26gt; 1) = 1 - P(X ≤ 1)


= 1 - P(X = 0) + P(X = 1)


= 1 - 0.857375 - 0.135375


= 0.00725


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