Example: Analyzing distribution of sum of two normally distributed random variables

shinji commutes to work and he worries about running out of fuel the amount of fuel he uses follows the normal distribution for each part of his commute but the amount of fuel he uses on the way home varies more the amount of fuel he uses for each part of the commute are also independent of each other here are summary statistics for the amount of fuel Shinji uses for each part of his commute so when he goes to work he uses a mean of 10 litres of fuel with a standard deviation of 1.5 liters and on the way home he also has a mean of 10 litres but there is more variation there's more spread he has a standard deviation of 2 litres suppose that shinji has 25 litres of fuel in his tank and he intends to drive to work and back home what is the probability that shinji runs out of fuel all right this is really interesting we have the distribution for the amount of fuel uses to work and to home and they say that these are normal distributions they say that right over here follows a normal distribution but here we're talking about the total amount of fuel he has to go to work and to go home so what we want to do is come up with a total distribution home and back I guess you could say we get to call this work + home home and back if you have two random variables that can be described by normal distributions and you were to define a new random variable as their sum the distribution of that new random variable will still be a normal distribution and its mean will be the sum of the means of those other random variables so the mean here I'll say the mean of work + home is going to be equal to 20 litres he will use a mean of 20 litres in the round trip now for the standard deviation from home + work you can't just add the standard deviations going and coming back but because the amount of fuel going to work and the amount of fuel coming home are independent random variables because they are independent of each other we can add the variances and only because they are independent can we add the variances so what you can say is that the variance of the combined trip is equal to the variance of going to work plus the variance of going home so what's the variance of going to work well 1.5 squared is so this will be one point five squared and what's the variance coming home well this is going to be two squared 2 squared well this is 2.25 plus 4 which is equal to 6 point 2 5 so the variance on the round trip is equal to 6 point 2 5 if I were to take the square root of that which is equal to 2 point 5 we can now describe the normal distribution of the round trip and use that to answer the question so we have this normal distribution that might look something like this we know it's mean is 20 liters so this is 20 liters and we want to know what is the probability that shinji runs out of fuel well to run out of fuel he would need to require more than 25 litres of fuel so if 25 litres of fuel is right over here so this is 25 litres of fuel the scenario where Shinji runs out of fuel is right over here this is where he needs more than 25 litres he actually has 25 litres in this tank so how do we figure out that area right over there well we could use a Z table we could say how many standard deviations above the mean is 25 liters well it is 5 litres above the mean so let me write this down so the Z here the Z is equal to 25 minus the mean minus 20 divided by the standard deviation for I guess you could say this combined normal distribution this is 2 standard deviations above the mean or z-score of +2 so if we look at a Z table and we look exactly two standard deviations above the mean that will give us this area the cumulative area below two standard deviations above the mean and then if we subtract that from 1 we will get the area that we care about so let's get our Z table out we care about a z-score of exactly two so two point zero zero is right over here Oh point nine seven seven two so that tells us that this area right over here is zero point nine seven seven two and so that blue area the probability that shinji runs out of fuel is going to be one minus zero point nine seven seven two and what is that going to be equal to let's see this is going to be equal to zero point zero two two eight did I do that right I think I did that right yes zero point two zero two two eight is the probability that shinji runs out of fuel if you want to think of it as a percent two point two eight percent chance that he runs out of fuel