Question

Three tables listed below show random variables and their probabilities. However, only one of these is actually a probability distribution.

A			B			C
x	P(x)		x	P(x)		x	P(x)
25	0.1		25	0.1		25	0.1
50	0.3		50	0.3		50	0.3
75	0.4		75	0.4		75	0.4
100	0.2		100	0.4		100	0.6

a. Which of the above tables is a probability distribution?

(Click to select)BAC

b. Using the correct probability distribution, find the probability that x is: (Round the final answers to 1 decimal place.)

1.	Exactly 75 =
2.	No more than 50 =
3.	More than 25 =

c. Compute the mean, variance, and standard deviation of this distribution. (Round the final answers to 2 decimal places.)

1.	Mean µ
2.	Variance σ2
3.	Standard deviation σ

Answer 1

a) To be a valid probability distribution, the sum of all proability must be equal to 1.

Sum for A = 0.1 + 0.3 + 0.4 + 0.2 = 1

Sum for B = 0.1 + 0.3 + 0.4 + 0.4 = 1.2

Sum for C = 0.1 + 0.3 + 0.4 + 0.6 = 1.4

So, Option-A) A is the correct answer.

b) P(exactly 75) = 0.4

P(no more than 50) = P(25) + P(50) = 0.1 + 0.3 = 0.4

P(more than 25) = 1 - P(25) = 1 - 0.1 = 0.9

c) Mean = E(X) = 25 * 0.1 + 50 * 0.3 + 75 * 0.4 + 100 * 0.2 = 67.5

E(X²) = 25² * 0.1 + 50² * 0.3 + 75² * 0.4 + 100² * 0.2 = 5062.5

Variance = E(X²) - (E(X))² = 5062.5 - 67.5² = 506.25

Standard deviation = sqrt(506.25) = 22.5