Video: Determining the Standard Deviation for a Discrete Random Variable

The function in the given table is a probability function of a discrete random variable 𝑋. Find the standard deviation of 𝑋. Give your answer to 2 decimal places.

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Video Transcript

The function in the given table is a probability function of a discrete random variable 𝑋. Find the standard deviation of 𝑋. Give your answer to two decimal places. When 𝑥 equals negative five, 𝑓 of 𝑥 equals a third. When 𝑥 is negative four, 𝑓 of 𝑥 is one- eighth. When 𝑥 is equal to negative three, 𝑓 of 𝑥 equals a quarter. And finally, when 𝑥 equals negative one, 𝑓 of 𝑥 is seven 24hs.

In order to calculate the standard deviation, we need to go through four steps. Firstly, we’ll calculate 𝐸 of 𝑥. This is the sum of 𝑥 multiplied by 𝑓 of 𝑥. Secondly, we’ll calculate 𝐸 of 𝑥 squared. This is the sum of 𝑥 squared multiplied by 𝑓 of 𝑥. Thirdly, we’ll calculate the Var of 𝑥 or variance of 𝑥. This is 𝐸 of 𝑥 squared minus the 𝐸 of 𝑥 all squared. And finally, to work out the standard deviation, we’ll square root our answer for the variance of 𝑥.

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 24hs. This is equal to negative 77 24hs.

The second step was to calculate 𝐸 of 𝑥 squared. 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. Well, negative five multiplied by negative five is 25. Negative four multiplied by negative four is positive 16. Negative three multiplied by negative three is nine. And finally, negative one multiplied by negative one is one.

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 24hs. This is equal to 103 eighths. Therefore, 𝐸 of 𝑥 squared is equal to a 103 eighths.

Our next step was to work out the variance of 𝑥. 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 24hs squared. This is equal to 1487 576 or 1487 divided by 576.

Our last step to calculate the standard deviation was to square root this answer. This is equal to 1.61 to two decimal places. Therefore, the standard deviation of the functions in the table is 1.61.

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