Question

The pmf for X = the number of major defects on a randomly selected gas stove of a certain type is

x 0 1 2 3 4
P(x) .10 .15 .45 .25 .05
Compute the following:
a. E(X)
b. V(X)
c. The standard deviation of

Answer 1

The expected value (E(X)) is 2.00, the variance (V(X)) is 1.00, and the standard deviation is 1.00.

Step-by-step explanation:

To calculate the expected value or mean of a discrete random variable, we multiply each value by its corresponding probability and then sum the products. For this problem, we have:

1. 0 * 0.10 = 0.00
2. 1 * 0.15 = 0.15
3. 2 * 0.45 = 0.90
4. 3 * 0.25 = 0.75
5. 4 * 0.05 = 0.20

Adding up these values gives us a mean of E(X) = 0.00 + 0.15 + 0.90 + 0.75 + 0.20 = 2.00.

The variance of a discrete random variable is calculated by subtracting the mean from each value, squaring the result, multiplying by the corresponding probability, and then summing the products. Using the mean of 2.00, we have:

1. (0 - 2)^2 * 0.10 = 4.00 * 0.10 = 0.40
2. (1 - 2)^2 * 0.15 = 1.00 * 0.15 = 0.15
3. (2 - 2)^2 * 0.45 = 0.00 * 0.45 = 0.00
4. (3 - 2)^2 * 0.25 = 1.00 * 0.25 = 0.25
5. (4 - 2)^2 * 0.05 = 4.00 * 0.05 = 0.20

Adding up these values gives us a variance of V(X) = 0.40 + 0.15 + 0.00 + 0.25 + 0.20 = 1.00.

The standard deviation is the square root of the variance, so for this problem, the standard deviation is √1.00 = 1.00.