Continuous Random Variables: Mean & Variance

this is a video about finding the mean and variance of a continuous random variable now the mean is the same as the expected value of the random variable and you may remember that for a discrete random variable that's equal to the sum of the probabilities times the possible values of the random variable now for a continuous random variable we replace the sum with an integral sign and we replace the probabilities with the probability density function so the mean of a continuous random variable is the integral of x f of x over all possible values of x the variance of a random variable which can be written Sigma squared or var X you may remember is equal to the expected value of x squared take away the square of the mean and for a discrete random variable that's equal to the sum of the probabilities times the squares of the possible values of X take away the mean squared and again for a continuous random variable you replace the sum with an integral sign and we replace the probabilities with the probability density function so the variance of a continuous random variable is the integral of x squared f of X evaluated over all possible values of X minus the square of the mean and these are the two things that you need to know to find the mean and the variance of a continuous random variable you need to know that the mean is the integral of x f of x over all possible values of x and the variance is the integral of x squared f of X over all possible values of X take away the square of the mean okay now let's look at some examples first of all suppose that we've got a random variable with the probability density function given by f of X is 3/10 of 3x minus x squared between 0 & 2 and 0 otherwise let's find a mean and a variance of X well to find the mean we have to integrate X f of X over all possible values of X and that's going to be the integral between 0 & 2 of x times 3/10 of 3x minus x squared we integrate between 0 & 2 because f of X is 0 everywhere else we can integrate this by multiplying out the brackets that gives us the integral of nine-tenths of x squared minus 3/10 of X cubed evaluated between Naughton 2 which is 3/10 of X cubed minus three fourteenth's of X to the power 4 evaluated between naught and 2 which is 3/10 of 2 cubed minus 3 40th sub 2 to the power 4 take away nothing which is 1 and 1/5 so that's the mean of this random variable secondly in order to find the variance of the random variable we have to calculate the expected value of x squared which is the integral of x squared f of X evaluated over all possible values of X and in order to do that we integrate x squared times 3/10 of 3x minus x squared between naught and 2 that's the integral between norton two of 9/10 of x cubed minus 3/10 of x to the power of 4 which is 9 fortieths of x to the power of 4 take away three fifths of X to power five evaluated between norton - in other words 948 x to the power 4 minus 350 s times 2 to the power 5 which is 42 over 25 ok well that's the value of the expected value of x squared but remember that in order to find a variance we have to take the expected value of x squared and subtract the square of the mean so the variance is going to be 42 over 25 take away the square of 6 fifths remember that the mean was 6 fifths so the answer here is that the variance is 6 over 25 okay now let's look at another example let's look at the random variable whose probability density function is given by f of X is 1/4 of X when X is between Naughton to 1/4 of 4 minus X when X is between 2 & 4 and 0 otherwise and let's find the mean and the standard deviation of X now we could work out the mean using integration but if you think about what the graph looks like there's a simpler way of doing it this time here's the graph of the probability density function and the first thing you'll notice is that it's symmetrical because it's symmetrical we can see straight away that the mean must be 2 okay so the mean for this random variable is 2 now let's work out the variance in order to find the variance we have to find the expected value of x squared and that would be the integral of x squared f of x over all possible values of X now to work that out we'll have to integrate x squared times 1/4 of X between 0 & 2 and then add the integral of x squared times 1/4 of 4 minus X between 2 & 4 this is because f of X is a piecewise defined function with one definition for when X is between 0 & 2 and a separate definition for when X is between 2 & 4 okay so we have to evaluate these two integrals and add them up let's tackle the first one let's work out the integral of x squared times 1/4 of X between 0 & 2 that's the same as the integral of 1/4 of X cubed between 0 & 2 which is going to be 1/16 of X to the power 4 evaluated between 0 & 2 which is 16 times 2 to the power 4 take away 0 which turns out to be 1 the other integral that we need to do is the integral of x squared times 1/4 of 4 minus X evaluated between 2 & 4 and that's the integral of x squared minus 1/4 of X cubed evaluate between two and four which is a third of x-cubed minus the sixteenth of x to the power of four between two and four which if you work it out turns out to be eleven thirds so now we know the value of each of these integrals we know that the first one is one and the second one is eleven-thirds so we can say that the expected value of x squared is equal to one plus eleven thirds which is fourteen thirds but remember it's terribly important that the variance is equal to the expected value of x squared take away the square of the mean so that's going to be fourteen thirds take away two squared because two is the mean and that's equal to two-thirds now this is the variance and the question asked us for the standard deviation we find the standard deviation by square rooting the variance and so it's equal to the square root of two-thirds which is not point eight one six two three significant figures okay let's look at one more example this is about the same random variable as before but the same probability density function but this time I'm asking you to calculate the probability that X is less than the mean take away the standard deviation well remember that the mean was two and the standard deviation was the square root of two-thirds so the mean take away the standard deviation will be two take away the square root of two-thirds which is about one point one eight three five zero three four so the question is asking us to find the probability that X is less than two take away the square root of two-thirds which is approximately the probability that X is less than one point point eight three five zero three four now there are different ways to work out this probability one way would be to integrate the function between zero and one point one eight 3503 four but if you remember that the graph of the probability density function is Lin it's made up out of straight lines is probably easier to work out the area of some basic geometric shapes we have to work out the area of this triangle and as the area of a triangle is 1/2 base times height the area is going to be 1/2 times one point one eight three five a 3/4 because that's the base times a quarter times one point one eight three five three four because that's the height better and more accurately we can say that the areas are 1/2 times 2 minus the square root of two thirds which is the base times 1/4 times two minus the square root of two thirds which is the height in other words the area's 1/8 of the square of to take away the square root of 2/3 which is no point 1 7 5 2 3 significant figures one last question on the same example this time let's calculate the probability that the absolute value of x take away the mean is less than the standard deviation and I want to look at this example because questions of this sort come up quite frequently in a-level exams now remember that the mean is 2 and the standard deviation is the square root of 2/3 so what the question is asking you is to find the probability that the absolute value of x minus 2 is less than the square root of 2/3 before we go any further though I'd like to look at what it means to take the absolute value of a minus B the modulus of a minus B and I want to think of it in terms of a number line there are two cases one possibility is that a is more than B for example a could be 8 and B could be 5 in that case the modulus of a minus B will be the modulus of 8 take away 5 which is the modulus of 3 which is 3 the other possibility is that B is greater than a for example B could be 8 and a could be 5 in that case the absolute value of a minus B is the absolute value of 5 minus 8 which is the absolute value of minus three and that's still equal to three and hopefully what you've noticed is that in both cases the modulus of a minus B or the absolute value of a minus B is the distance between a and B on the number line so now let's look at the probability that we're supposed to work out again we're supposed to work out the probability that the absolute value of X minus 2 is less than the square root of 2/3 now what this inequality is saying is that the distance between X and 2 is less than the square root of 2/3 X is no further away from 2 than the square root of 2/3 or in other words it's greater than 2 minus the square root of 2/3 and it's less than 2 plus the square root of 2/3 so the probability that we're looking for is the probability that X is greater than 2 less a little bit and less than 2 plus a little bit it's probably easier to imagine this as an area on the graph so we're trying to find this area the region where X is greater than 2 take away the square root of 2/3 and X is less than 2 plus the square root of 2/3 now there are various ways to work out this area perhaps the simplest however is to realize that there's a connection between the red area and the yellow area here the red area is going to be one takeaway the yellow area because the total area of the triangle must be one the total area under a probability density function has to be one so that means that the red area is about one takeaway twice the probability that X is less than one point one eight three five zero three four remember that the area of one yellow triangle is the probability that X is less than one point one eight 3503 for but there are two yellow triangles which have the same area using the answer that we got earlier we can say that this is about one take away two times 0.175 508 504 okay well these calculations will give us the approximate answers but we can probably do a little bit better by thinking in terms of SERDES we can say that the red area will be one take away two times the probability that X is less than two minus the square root of two-thirds and again using the answer that we got earlier that's going to be one take away two times and eight that the square of two minus the square root of two-thirds which is one take away a quarter of the square of two minus the square root of two-thirds which is not 0.65 Oh two three significant figures okay this is the end of my video about finding the mean and the variance of a continuous random variable the main things that you need to remember that the mean is given by the integral of X f of X over all possible values of X and the variance is given by the integral of x squared f of X evaluated over all possible values of X take away the square of the mean okay I hope that you found this video useful thank you very much for watching

Channel: MrNichollTV

Views: 218,250

Keywords: Mathematics, Statistics, A-Level, S2, Continuous Random Variables, Mean, Variance, Expectation

Id: gPAxuMKZ-w8

Length: 14min 31sec (871 seconds)

Published: Wed Dec 05 2012