Calculating the Standard Deviation of a Random Variable

Given the probability distribution of a random variable X, we can calculate the standard deviation of X using the following formula:

σ = √(Σ(xi - μ)²P(xi))

where:
- σ is the standard deviation
- μ is the mean or expected value of X
- xi is each possible value of X
- P(xi) is the probability of X taking on the value xi

Steps:
1. Calculate the mean or expected value of X.
2. Calculate the variance of X using the formula: Var(X) = Σ(xi - μ)²P(xi)
3. Take the square root of the variance to get the standard deviation: σ = √Var(X)

Calculating the Standard Deviation of the Given Probability Distribution

Given:
X: 1 2 4 5 6
P: 0.15 0.25 0.2 0.3 0.1

1. Calculate the mean or expected value of X.
μ = ΣxiP(xi) = (1)(0.15) + (2)(0.25) + (4)(0.2) + (5)(0.3) + (6)(0.1) = 3.55

2. Calculate the variance of X using the formula: Var(X) = Σ(xi - μ)²P(xi)
Var(X) = (1 - 3.55)²(0.15) + (2 - 3.55)²(0.25) + (4 - 3.55)²(0.2) + (5 - 3.55)²(0.3) + (6 - 3.55)²(0.1) = 2.3025

3. Take the square root of the variance to get the standard deviation: σ = √Var(X) = √2.3025 = 1.5165

Therefore, the standard deviation of X is 1.5165.