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Expected Value

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Hi. This tutorial covers expected value. OK, so let's start with a situation, specifically a chance experiment. So the chance experiment here is to note the gender of the last three babies born at a local hospital. OK, so one outcome might be male, female, female.

And what we're going to do is we're going to let the random variable X represent the number of girls among the three babies observed. We're looking specifically for the number of girls. So if our outcome was male, female, female, X would equal 2 for that trial.

OK, so now let's consider the discrete probability distribution of X. So the possible number of girls out of three babies being born are 0, 1, 2, or 3. So it could be all boys, so no girls, 1 girl, 2 boys, 2 girls, 1 boy, 3 girls and no boys. OK, so these probabilities were calculated using kind of a 50% chance of a boy, 50% chance of a girl.

OK, so now the question is if three babies are born, how many would we expect to be girls? Would we expect 1 out of 3 to be girls? Would we expect 2 out of 3 to be girls? OK, so what we want to be able to do is use a probability distribution to calculate what we call the expected value.

So to define expected value, it's the weighted average of all possible values of the random variable. It's also known as the mean of a probability distribution, and the mean of the probability distribution we use the symbol mu. Remember that mu represents a population mean, but since we're doing it for a probability distribution, this is what we would expect the population mean to be. So we're going to use that Greek letter mu.

So then kind of putting this into a formula, the formula for the expected value of X, so sometimes it's notated as E of X, where E represents E for expected value. So the expected value of X is the sum of X times P of X. So you're going to take each possible value of X, multiply it by its corresponding probability, and then add all of those up.

And then this is the same formula down here, but we're just using that Greek letter mu, so the average, the population average, or the expected values. The sum of X times p of X. And anytime you have the sum symbol here using that summation notation, there's always implied parentheses after that. So we're always going to multiply first and then add at the end.

OK, so let's take a look now at actually calculating the expected value using this probability distribution. So again, the formula is the sum of X times P of X. OK, so sometimes what I like to do is just add in kind of a third column to this table and then just write down what X times P of X is, because all we're going to do is multiply X times P of X, X times P of X, X times P of X.

OK, let's do that on the calculator, and I'll write down the values as we go. So let's start with 0 times 0.125, which is obviously 0. 0 times anything is 0. So I'm going to put a 0 here. My next one is 1 times 0.375. Anything times 1 is itself. So that's going to be 0.375 there.

OK, the next one is going to be 2 times 0.375, and that ends up being 0.75. So that's what I'll put here. And the last one is 3 times 1.25. So 3 times-- or sorry, 0.125, and that's 0.375.

OK, now what we want is we want the sum of X times P of X. So we need the sum of these values. So I need to add those up. OK, so now let's add those four values in the calculator. So 0 plus 0.375 plus 0.75 plus 0.375. OK, so those are those four values all added, and I end up with a mean or an expected value of 1.5.

So for three newborn babies, 1.5 females are expected. OK, so 1.5 is the expected value. OK, now can you have 1.5 females from a sample of 3? Obviously, no. When you're actually finding values of X, you can only have whole numbers. But it is OK to have an expected value that's like a decimal number. And the reason behind that is is this is just what you would expect for 3.

OK, if you had 6 newborn babies, we would expect 3 females. So even though 1.5 isn't possible, it's still a meaningful value. All right, so the key on this tutorial was just to make sure you understand how to apply the expected value formula or the mean. So that has been your tutorial on the expected value. Thanks for watching.