Example: Analyzing distribution of sum of two normally distributed random variables | Khan Academy

Shinji commutes to work, and he worries about running out of fuel. The amount of fuel he uses follows a normal distribution for each part of his commute, but the amount of fuel he uses on the way home varies more. The amounts 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: when he goes to work, he uses a mean of 10 L of fuel with a standard deviation of 1.5 L. On the way home, he also has a mean of 10 L, but there is more variation; there is more spread, and he has a standard deviation of 2 L.

Suppose that Shinji has 25 L 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 distributions for the amount of fuel he 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 could call this work plus 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. 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 plus home, is going to be equal to 20 L. He will use a mean of 20 L in the round trip. Now, for the standard deviation from home plus 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.

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 1.5 squared. And what's the variance coming home? Well, this is going to be 2 squared. Well, this is 2.25 + 4, which is equal to 6.25. So, the variance on the round trip is equal to 6.25. If I were to take the square root of that, which is equal to 2.5, 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 its mean is 20 L, so this is 20 L. 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 L of fuel. So, if 25 L of fuel is right over here, so this is 25 L of fuel, the scenario where Shinji runs out of fuel is right over here. This is where he needs more than 25 liters; he actually has 25 L in his 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 L. Well, it is 5 L 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 two standard deviations above the mean or a Z score of plus two.

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. Then, if we subtract that from one, 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 2. is right over here, 0.9772. So, that tells us that this area right over here is 0.9772.

So, that blue area, the probability that Shinji runs out of fuel is going to be 1 minus 0.9772. And what is that going to be equal to? Let's see, this is going to be equal to 0.0228. Did I do that right? I think I did that right, yes. 0.0228 is the probability that Shinji runs out of fuel. If you want to think of it as a percent, 2.28% chance that he runs out of fuel.