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	0.6		50	0.6		50	0.6
75	0.1		75	0.1		75	0.1
100	0.2		100	0.4		100	0.6

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

(Click to select)  A  B  C

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 75 =
3.	More than 75 =

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

Part a

Which of the above tables is a probability distribution?

Answer: A

(Because the sum of probabilities for a probability distribution in table A is 1, so this is a probability distribution. For table B and C, the sum of probabilities is not equal to 1.)

Part b

Required probabilities are given as below:

1.	Exactly 75 = P(X=75) = 0.1	Answer: 0.1
2.	No more than 75 = P(X=25) + P(X=50) + P(X=75) = 0.1+0.6+0.1 = 0.8	Answer: 0.8
3.	More than 75 = P(X=100) = 0.2	Answer: 0.2

Part c

Formulas for mean, variance, and standard deviation are given as below:

Mean = µ = ∑X*P(X)

Variance = σ² = ∑(X - µ)²*P(X)

Standard deviation = σ = sqrt[∑(X - µ)²*P(X)]

Calculation table is given as below:

X	P(X)	XP(X)	(X - mean)^2	(X - mean)^2*P(X)
25	0.1	2.5	1225	122.5
50	0.6	30	100	60
75	0.1	7.5	225	22.5
100	0.2	20	1600	320
Total	1	60		525

Mean = µ = ∑X*P(X) = 60

Variance = σ² = ∑(X - µ)²*P(X) = 525

Standard deviation = σ = sqrt[∑(X - µ)²*P(X)] = sqrt(525) = 22.91287847

1.	Mean µ	60.00
2.	Variance σ2	525.00
3.	Standard deviation σ	22.91