Question

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

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

(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 =

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