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