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Let π denote a discrete random variable which can take the values negative one, π, and one.
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Given that π has probability distribution function π of π₯ equals π₯ plus two all over six, find the expected value of π.
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A probability distribution is usually given as a table of values, showing the probabilities of various outcomes of an experiment.
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It will be hugely useful then to convert the information in this question into table form.
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We know that the discrete random variable π₯ can take the values negative one, π, and one.
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To find their associated probabilities, we, therefore, need to substitute these values of π₯ into our function.
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π of negative one is negative one plus two all over six, which is a sixth.
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π of π is π plus two all over six.
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And π of one is one plus two all over six, which is three-sixths.
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Now, weβre going to leave that as three-sixths rather than a half for reasons that will become clear in a moment.
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Now, remember because these are all the possible outcomes of an experiment, the sum of the probabilities is one.
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We can, therefore, form an equation in terms of π by summing these probabilities and making them equal to one.
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Simplifying fully the left-hand side, we get π plus six all over six.
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Notice how weβve changed one into six-sixths as it allows us to simplify further our equation.
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Because the denominators are the same, the numerators must also be equivalent.
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π plus six is equal to six.
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And we can, therefore, subtract six from both sides to get π is equal to zero.
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We can then change the value of π to zero in our table and the probability is two-sixths.
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Now, remember to find the expected value of π, we add together the product of each discrete random variable and its associated probability.
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Thatβs negative one multiplied by a sixth plus zero multiplied by two-sixths plus one multiplied by three-sixths.
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Simplifying, we get negative a sixth from the first bracket, zero from the second, and three-sixths from the third, which gives us two-sixths.
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This simplifies finally to a third.
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The expected value of π is a third.