Consider the probability distribution below. The variance of x equals
A) 9.
B) 23.
C) 81.
D) 610.

Question

Consider the probability distribution below.  ​ The variance of x equals
A) 9. 
B) 23. 
C) 81. 
D) 610.

Quizplus · Accepted Answer

To find the variance of x, we need to first find the expected value of x, denoted E(x).

E(x) = Σ(xi * P(xi)) for all i from 1 to 6

E(x) = (2*0.17) + (3*0.26) + (4*0.19) + (5*0.12) + (6*0.16) + (7*0.10) = 4.19

Next, we need to find the squared deviations from the expected value for each possible outcome:

(2-4.19)² * 0.17 = 4.36 
(3-4.19)² * 0.26 = 0.79 
(4-4.19)² * 0.19 = 0.03 
(5-4.19)² * 0.12 = 0.37 
(6-4.19)² * 0.16 = 2.02 
(7-4.19)² * 0.10 = 5.61

The variance is the sum of these squared deviations multiplied by their probabilities:

Variance = Σ[(xi - E(x))² * P(xi)] for all i from 1 to 6

Variance = 4.36 + 0.79 + 0.03 + 0.37 + 2.02 + 5.61 = 13.18

Thus, the variance of x is 13.18. To check our work, we can take the square root of the variance to find the standard deviation:

Standard deviation = sqrt(variance) = sqrt(13.18) ≈ 3.63

Note: Choice C, 81, is the square of the standard deviation, which is a common mistake. The correct answer is the variance, which is not squared.

Consider the Probability Distribution Below