Determine the mean and standard deviation of the random variable for which the probability mass function is defined as follows:
P(X=x)= x-2/30 if x=3,12,21
To determine the mean and standard deviation of a random variable with a probability mass function (PMF), follow these steps:
Step 1: Calculate the expected value (mean).
The mean of a discrete random variable is calculated by multiplying each possible value by its corresponding probability and summing them up. In this case, the PMF is as follows:
P(X=3) = (3-2)/30 = 1/30
P(X=12) = (12-2)/30 = 10/30
P(X=21) = (21-2)/30 = 19/30
To calculate the mean, multiply each value by its corresponding probability and sum them up:
Mean (μ) = 3 * P(X=3) + 12 * P(X=12) + 21 * P(X=21)
= 3 * (1/30) + 12 * (10/30) + 21 * (19/30)
= (3/30) + (120/30) + (399/30)
= 522/30
= 17.4
So, the mean of the random variable is 17.4.
Step 2: Calculate the standard deviation (σ).
The standard deviation of a discrete random variable can be obtained using the following formula:
σ = √((Σ(x - μ)^2 * P(x))
First, calculate the squared difference between each value and the mean, multiplied by their respective probabilities:
(3 - 17.4)^2 * (1/30) = 2214/30
(12 - 17.4)^2 * (10/30) = 108/30
(21 - 17.4)^2 * (19/30) = 2574/30
Next, sum up these values:
Σ((x - μ)^2 * P(x)) = 2214/30 + 108/30 + 2574/30
= 4896/30
= 163.2
Finally, take the square root of this result to obtain the standard deviation:
σ = √(163.2)
= 12.77
So, the standard deviation of the random variable is approximately 12.77.