WEBVTT
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The function in the given table is a probability function of a discrete random variable 𝑋.
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Find the standard deviation of 𝑋.
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Give your answer to two decimal places.
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The table tells us that the probability that 𝑥 equals four is eight 19ths.
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The probability that 𝑥 equals six is also eight 19ths.
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And the probability that 𝑥 equals 10 is three 19ths.
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In order to answer this question, we need to follow four steps.
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Firstly, we need to calculate 𝐸 of 𝑥.
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Secondly, 𝐸 of 𝑥 squared.
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Thirdly, we will use these two answers to calculate the variance of 𝑥.
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And finally we’ll calculate the standard deviation of 𝑥.
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𝐸 of 𝑥 is the sum of all the 𝑥-values multiplied by the 𝐹 of 𝑥 values, in this case four multiplied by eight 19ths plus six multiplied by eight 19ths plus 10 multiplied by three 19ths.
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This gives us a 110 19ths, or 110 divided by 19.
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𝐸 of 𝑥 equals 110 divided by 19.
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Our second step is to calculate 𝐸 of 𝑥 squared.
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𝐸 of 𝑥 squared is equal to the sum of the 𝑥 squared values multiplied by 𝐹 of 𝑥, in this case four squared or 16 multiplied by eight 19ths, six squared multiplied by eight 19ths, and 10 squared multiplied by three 19ths.
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The sum of these is 716 19ths.
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Therefore, 𝐸 of 𝑥 squared is equal to 716 divided by 19.
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Our third step is to work out the variance of 𝑥.
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This is calculated by subtracting the 𝐸 of 𝑥 all squared from the 𝐸 of 𝑥 squared, in this case 716 19ths minus a 110 19ths squared.
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Therefore, the variance is 1504 divided by 361, or 4.166.
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Our final step is to calculate the standard deviation.
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This is calculated by square-rooting the variance.
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The square root of 1504 divided by 361 is equal to 2.04.
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Therefore, the standard deviation of the function in the table is 2.04.