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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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When 𝑥 equals negative five, 𝑓 of 𝑥 equals a third.
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When 𝑥 is negative four, 𝑓 of 𝑥 is one- eighth.
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When 𝑥 is equal to negative three, 𝑓 of 𝑥 equals a quarter.
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And finally, when 𝑥 equals negative one, 𝑓 of 𝑥 is seven 24ths.
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In order to calculate the standard deviation, we need to go through four steps.
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Firstly, we’ll calculate 𝐸 of 𝑥.
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This is the sum of 𝑥 multiplied by 𝑓 of 𝑥.
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Secondly, we’ll calculate 𝐸 of 𝑥 squared.
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This is the sum of 𝑥 squared multiplied by 𝑓 of 𝑥.
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Thirdly, we’ll calculate the Var of 𝑥 or variance of 𝑥.
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This is 𝐸 of 𝑥 squared minus the 𝐸 of 𝑥 all squared.
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And finally, to work out the standard deviation, we’ll square root our answer for the variance of 𝑥.
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The 𝐸 of 𝑥 was the sum of 𝑥 multiplied by 𝑓 of 𝑥 — in this case negative five multiplied by a third plus negative four multiplied by an eighth plus negative three multiplied by a quarter plus negative one multiplied by seven 24ths.
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This is equal to negative 77 24ths.
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The second step was to calculate 𝐸 of 𝑥 squared.
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In order to do this, we’ll firstly gonna calculate the 𝑥 squared values: negative five squared, negative four squared, negative three squared, and negative one squared.
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Well, negative five multiplied by negative five is 25.
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Negative four multiplied by negative four is positive 16.
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Negative three multiplied by negative three is nine.
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And finally, negative one multiplied by negative one is one.
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This means that 𝐸 of 𝑥 squared is equal to 25 multiplied by a third plus 16 multiplied by an eighth plus nine multiplied by a quarter plus one multiplied by seven 24ths.
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This is equal to 103 eighths.
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Therefore, 𝐸 of 𝑥 squared is equal to a 103 eighths.
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Our next step was to work out the variance of 𝑥.
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This was calculated by subtracting the 𝐸 of 𝑥 all squared from the 𝐸 of 𝑥 squared — in this case 103 eighths minus negative 77 24ths squared.
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This is equal to 1487 576 or 1487 divided by 576.
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Our last step to calculate the standard deviation was to square root this answer.
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This is equal to 1.61 to two decimal places.
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Therefore, the standard deviation of the functions in the table is 1.61.