Video: Calculations Involving Expected Values

An experiment that produces the discrete random variable 𝑋 has the probability distribution shown. Calculate 𝐸(𝑋). Calculate 𝐸(𝑋²). The variance of 𝑋 can be calculated using the formula Var(𝑋) = 𝐸(𝑋²) βˆ’ 𝐸(𝑋)Β². Calculate Var(𝑋) to 2 decimal places.

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Video Transcript

An experiment that produces the discrete random variable 𝑋 has the probability distribution shown. Calculate 𝐸 of 𝑋, the expected value of 𝑋.

This value here is the expected value of 𝑋, which we have a formula for. It’s the sum of the products of the values of 𝑋 with the probability that 𝑋 takes that value. So in our problem, 𝐸 of 𝑋 is two times the probability that 𝑋 is two, 0.1, plus three times the probability that 𝑋 is three, that is 0.3, plus four times 0.2 plus five times 0.4. Each possible outcome of 𝑋 contributes a term to this sum. Evaluating this sum, we get 3.9. This value is the expected value of 𝑋 because it gives kind of the average value of 𝑋 that we would expect. If we repeated the experiment 𝑛 times, the sum of our outcomes would be around 3.9 times 𝑛.

The next part of our question is to calculate the expected value of 𝑋 squared.

This quantity tells us what we should expect the square of the outcome of the experiments to be on average. It’s important to note that the expected value of 𝑋 squared is not the same thing as the expected value of 𝑋, squared. We add the definition of the expected value of 𝑋 squared to the definition of the expected value of 𝑋. We use the definition of the expected value of 𝑋 squared to compute this for our example. The first outcome, which is two, contributes two squared times 0.1. The second contributes three squared times 0.3. We add four squared times 0.2 and five squared times 0.4. Putting this into our calculators, we get 16.3. We can see that as claimed, this value is different from the expected value of 𝑋, which we found to be 3.9 squared.

The final part of this question is: The variance of 𝑋 can be calculated using the formula Var of 𝑋, or the variance of 𝑋, is equal to the expected value of 𝑋 squared minus the expected value of 𝑋, squared. Calculate Var of 𝑋, the variance of 𝑋, to two decimal places.

We’ve calculated the expected value of 𝑋 squared already. It’s 16.3. And from this, we have to subtract the expected value of 𝑋, squared. So that’s 3.9 squared. Putting this into our calculators, we get 1.09. There is therefore no need to round this value to two decimal places because it already has only two decimal places. And so this is our answer: the variance of 𝑋 is 1.09.

Well, the expected value of the discrete random variable 𝑋 gives you a kind of representative or average outcome of the random variable. The variance of 𝑋 tells you how spread out the outcomes are. To calculate the variance of a discrete random variable 𝑋, you not only need the expected value of 𝑋, you also need the expected value of 𝑋 squared. It’s important to know that this is not the same as the expected value of 𝑋, squared. If it were, then the variance of a discrete random variable 𝑋 would always be zero, from the definition. The variance of a discrete random variable 𝑋 is the second most important thing after it’s expected value. And so the variance will turn up a lot in high-level statistics.

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