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An experiment that produces the discrete random variable π has the probability distribution shown.
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Calculate πΈ of π, the expected value of π.
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This value here is the expected value of π, which we have a formula for.
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Itβs the sum of the products of the values of π with the probability that π takes that value.
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So in our problem, πΈ of π is two times the probability that π is two, 0.1, plus three times the probability that π is three, that is 0.3, plus four times 0.2 plus five times 0.4.
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Each possible outcome of π contributes a term to this sum.
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Evaluating this sum, we get 3.9.
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This value is the expected value of π because it gives kind of the average value of π that we would expect.
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If we repeated the experiment π times, the sum of our outcomes would be around 3.9 times π.
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The next part of our question is to calculate the expected value of π squared.
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This quantity tells us what we should expect the square of the outcome of the experiments to be on average.
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Itβs important to note that the expected value of π squared is not the same thing as the expected value of π, squared.
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We add the definition of the expected value of π squared to the definition of the expected value of π.
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We use the definition of the expected value of π squared to compute this for our example.
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The first outcome, which is two, contributes two squared times 0.1.
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The second contributes three squared times 0.3.
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We add four squared times 0.2 and five squared times 0.4.
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Putting this into our calculators, we get 16.3.
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We can see that as claimed, this value is different from the expected value of π, which we found to be 3.9 squared.
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The final part of this question is: The variance of π can be calculated using the formula Var of π, or the variance of π, is equal to the expected value of π squared minus the expected value of π, squared.
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Calculate Var of π, the variance of π, to two decimal places.
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Weβve calculated the expected value of π squared already.
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Itβs 16.3.
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And from this, we have to subtract the expected value of π, squared.
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So thatβs 3.9 squared.
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Putting this into our calculators, we get 1.09.
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There is therefore no need to round this value to two decimal places because it already has only two decimal places.
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And so this is our answer: the variance of π is 1.09.
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Well, the expected value of the discrete random variable π gives you a kind of representative or average outcome of the random variable.
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The variance of π tells you how spread out the outcomes are.
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To calculate the variance of a discrete random variable π, you not only need the expected value of π, you also need the expected value of π squared.
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Itβs important to know that this is not the same as the expected value of π, squared.
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If it were, then the variance of a discrete random variable π would always be zero, from the definition.
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The variance of a discrete random variable π is the second most important thing after itβs expected value.
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And so the variance will turn up a lot in high-level statistics.