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 0.3 25 0.3 25 0.3 50 0.1 50 0.1 50 0.1 75 0.2 75 0.2 75 0.2 100 0.4 100 0.6 100 0.8 a. Which of the above tables is a probability distribution? b. Using the correct probability distribution, find the probability that x is: (Round the final answers to 1 decimal place.) 1. Exactly 100 = 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

(a)

From the given data, the following Table is formed:

For A:

x	P(x)
25	0.3
50	0.1
75	0.2
100	0.4
Total =	1.0

For B:

x	P(x)
25	0.3
50	0.1
75	0.2
100	0.6
Total =	1.2

For C;

x	P(x)
25	0.3
50	0.1
75	0.2
100	0.8
Total =	1.4

It is noted that in case of A only, Total Probability = 1.

So,

Only A is actually a probability distribution.

(b)

(1)

P(x = 100) = 0.4

(2)
P(X75) = P(X=25) + P(X=50) + P(X=75) = 0.3 + 0.1 + 0.2 = 0.6

(3)
P(X>75) = 0.4

(c)

From the Probability Distribution, the following Table is formed:

x	P(x)	x p	x2 p
25	0.3	7.5	187.5
50	0.1	5	250
75	0.2	15	1125
100	0.4	40	4000
Total =	1.0	67.50	5562.50

(1) Mean = 67.50

(2) Variance = E(X²) - (E(X))² = 5562.50 - 67.50³ = 1006.25

(3) Standard Deviation =