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Finding the mean and standard deviation of a binomial random variable

Video transcript

we're told a company produces processing chips for cell phones at one of its large factories two percent of the chips produced are defective in some way a quality check involves randomly selecting and testing 500 chips what are the mean and standard deviation of the number of defective processing chips in these examples so like always try to pause this video and have a go at it on your own and then we will work through it together alright so let me define a random variable that we're gonna find the mean and standard deviation of and I'm gonna make that random variable the number of defective processing chips in a 500 chip sample so let's let X be equal to the number of defective chips in 500 chip sample so the first thing to recognize is that this will be a binomial variable this is binomial how do we know it's binomial well it's made up of 500 it's a finite number of trials right over here the probability of getting a defective chip you could view this as a probability of success it's a little bit counterintuitive that a defective chip would be a success but we're summing up the defective chips so we would view the probability of a defect or I should say a defective chip it is constant across these 500 trials and we will assume that they are independent of each other 0.02 you might be saying hey well are we replacing the chips before after but we're assuming that's from a functionally infinite population or if you want to make it feel better you could say well maybe you are replacing the chips they're not really telling us that right over here so we'll assume that each of these trials are independent of each other and that the probability of getting a defective chip stays constant here and so this is a binomial random variable or binomial variable and we know the formulas for the mean and standard deviation of a binomial variable so the mean the mean of X which is the same thing as the expected value of X is going to be equal to the number of trials n times the probability of a success on each trial times P so what is this going to be well this is going to be equal to we have 500 trials and then the probability of success on each of these trials is 0.02 so it's 500 times 0.02 and what is this going to be 500 times 2 hundredths is going to be it's going to be equal to 10 so that is your expected value of the number of defective processing chips or the mean now what about the standard deviation so the standard deviation of our random variable X well that's just going to be equal to the square root of the variance of our random variable X so I could just write it I'm just writing it all the different ways that you might see it because sometimes the notation is the most confusing part in statistics and so this is going to be the square root of what well the variance of a binomial variable is going to be equal to the number of trials times the probability of success in each trial times 1 minus the probability of success in each trial and so in this context this is going to be equal to you're gonna have the 500 500 times zero point zero two 0.02 times 1 minus 0.02 is 0.98 so times 0.98 and all of this is under the radical sign I didn't make that radical sign long enough and so what is this going to be well let's see 500 times zero point zero two we already said that this is going to be 10 ten times zero point nine eight this is going to be equal to the square root of nine point eight so it's going to be I don't know three point something if we want we can get a calculator out to feel a little bit better about this value so I'm gonna take nine point eight and then take the square root of it and I get three point five round to the nearest hundredth three point one three so this is approximately three point one three for the standard deviation if I wanted the variance it would be nine point eight but they asked for the standard deviation so that's why we got that alright hopefully you enjoyed that