Question

iven the probability density function f(x) = (1/3) over the interval [1, 4], find the expected value, the mean, the variance, and the standard deviation. Expected value: ______ Mean: ______ Variance: ______ Standard deviation: ______.

Answer 1

The expected value is 7/6, the mean is 7/6, the variance is 4912/27, and the standard deviation is (16√3)/9.

Step-by-step explanation:

To find the expected value, mean, variance, and standard deviation for the given probability density function, we can use the formulas:

Expected value: E(X) = ∑ (x * P(X = x))

Mean: μ = E(X)

Variance: Var(X) = ∑ ((x - μ)2 * P(X = x))

Standard deviation: σ = √(Var(X))

For the given probability density function f(x) = (1/3) over the interval [1, 4], the expected value, mean, variance, and standard deviation can be calculated as follows:

Expected value: E(X) = (∫ (x * f(x)) dx) = (∫ x(1/3) dx) = (1/3) * (∫ x dx over [1, 4]) = (1/3) * [(x2/2)] over [1, 4] = (1/3) * [(42/2) - (12/2)] = (1/3) * [(16/2) - (1/2)] = (1/3) * [(8 - 1)/2] = (1/3) * (7/2) = 7/6

Mean: μ = Expected value = 7/6

Variance: Var(X) = (∫ ((x - μ)2 * f(x)) dx) = (∫ ((x - 7/6)2 * (1/3)) dx over [1, 4]) = (1/3) * (∫ (x - 7/6)2 dx over [1, 4]) = (1/3) * ((1/3) * ((x - 7/6)3/3)) over [1, 4] = (1/3) * ((1/3) * ((4 - 7/6)3/3) - (1 - 7/6)3/3)) = (1/3) * ((1/3) * ((24 - 7)3/3) - (6 - 7)3/3)) = (1/3) * ((1/3) * (173/3) - (1/3)3/3) = (1/3) * ((1/3) * (4913/3) - (1/3)/27) = (1/3) * ((4913/9) - (1/27)) = (1/3) * ((4913/9) - (1/9)) = (1/3) * ((4913 - 1)/9) = (1/3) * (4912/9) = 4912/27

Standard deviation: σ = √(Var(X)) = √(4912/27) = √(4912)/√(27) = (16√3)/9