Geometric Distribution: Mean and Variance In Exercises 29 and ...

Question

Geometric Distribution: Mean and Variance In Exercises 29 and 30, use the fact that the mean of a geometric distribution is μ = 1/p and the variance is

sigma^2 = q/p^2

Paycheck Errors A company assumes that 0.5% of the paychecks for a year were calculated incorrectly. The company has 200 employees and examines the payroll records from one month. (a) Find the mean, variance, and standard deviation. (b) How many employee payroll records would you expect to examine before finding one with an error?

Accepted Answer

Welcome back, everyone. A factory finds that 1.2% of its products are defective. Suppose the products are inspected one at a time. A compute the mean, variance, and standard deviation for the number of inspections needed to find the first effective product. B, what is the expected number of inspections before finding a defective item. For this problem, let's understand that we're dealing with a geometric distribution because it says to find the first defective product, right? So we have a sequence of failures, and then finally we have a success. This is a geometric distribution. So we can begin with A and identify the mean using the mean formula for a geometric distribution. We simply divide 1 by the probability of success B. And the probability of success is 1.2% or 0.012 in its decimal form. Dividing the two numbers, we get 83.33 for the mean value. Now, let's identify the variants. The variance formula says that we have to take the complement of P 1 minus P. And divided by p squared. So that would be 1 minus 0.012 divided by 0.012 squared. Which is equal to 6,861.11. And then we can identify the standard deviation, which is simply square root of the variance. Or square root of 1 minus p divided by p squad. This is equal to square root of 1 minus 0.012. Divided by 0.012 squared. Which is equal to 82.83. Well then, so we have our 1st 3 answers, the mean value, the variance, and the standard deviation. We are ready to focus on Part B. It says what is the expected number of inspections before finding a defective item. So we have to recall that the expected number is. Of X is equal to the mean value. And we know that our mean is 83.33, right? And essentially what we can do is simply round it to the rounded to the nearest whole number. So that would be 83. The reason why we're rounding is because the number of inspections is a whole number, but essentially it is equal to our mean value. That will be our final answer, and thank you for watching.

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Key Concepts

Geometric Distribution

Mean and Variance

Standard Deviation

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