Question

For the following probability distribution:

X	3	4	5
P(X)	0.5	0.2	0.3

The mean, variance, and standard deviation respectively are:

• A. 4, 3.8, and 0.87
• B. 4, 3.8, and 0.76
• C. 3.8, 4, and 0.76
• D. 3.8, 0.76, and 0.87

Answer 1

The Correct Option is D

The process to determine the mean, variance, and standard deviation for the provided probability distribution involves the following steps:

1. Mean (\(\mu\)):
   The mean is computed as \(\mu = \sum (X \times P(X))\).
   Calculation:
   \(\mu = (3 \times 0.5) + (4 \times 0.2) + (5 \times 0.3) = 1.5 + 0.8 + 1.5 = 3.8\).
2. Variance (\(\sigma^2\)):
   Variance is calculated using the formula \(\sigma^2 = \sum ((X - \mu)^2 \times P(X))\).
   Calculation:
   \(\sigma^2 = ((3 - 3.8)^2 \times 0.5) + ((4 - 3.8)^2 \times 0.2) + ((5 - 3.8)^2 \times 0.3)\)
   \(\sigma^2 = (0.64 \times 0.5) + (0.04 \times 0.2) + (1.44 \times 0.3)\)
   \(\sigma^2 = 0.32 + 0.008 + 0.432 = 0.76\).
3. Standard Deviation (\(\sigma\)):
   The standard deviation is the square root of the variance:
   \(\sigma = \sqrt{0.76} \approx 0.87\).

Consequently, the mean, variance, and standard deviation are 3.8, 0.76, and 0.87, respectively.