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Work out the expected value of the random variable π whose probability distribution is shown.
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Letβs begin by presenting this information in table form.
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Itβs not necessary though it does make it easier to work out what we need to do.
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The four possible values for our random variable π₯ are one, two, three, and four.
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The probability that π is equal to one is given by the height of this first bar; itβs 0.2.
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The probability that π is equal to two is given by the height of the second bar, which is 0.3.
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The probability that π is equal to three is 0.3.
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And the probability that π is equal to four is 0.2.
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We can double-check whether weβve calculated these probabilities correctly because we know that they should all sum to one: 0.2 plus 0.3 plus 0.3 plus 0.2 does indeed equal one.
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Now, letβs recall the formula for the expected value of π.
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Itβs the sum of each of the possible outcomes multiplied by the probability of this outcome occurring.
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Letβs substitute then what we have into this formula.
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For the first column, π₯ multiplied by the probability of π is one multiplied by 0.2.
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For the second column, itβs two multiplied by 0.3.
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The probability that π is equal to three is 0.3.
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So π₯ multiplied by the probability of π here is three multiplied by 0.3.
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And for our fourth and final column, itβs four multiplied by 0.2.
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Evaluating each of these products, we get 0.2 plus 0.6 plus 0.9 plus 0.8 which is 2.5.
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So we have that the expected value of π is 2.5.
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We can look at our table to check whether this answer is likely to be correct.
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Since the possible values of π₯ are one, two, three, and four and 2.5 is halfway between one and four, 2.5 is likely to be correct for the expected value of our probability distribution.