Answer:
Mean waiting time = E(X) = 0 minute
Variance = Var(X) = 140
Step-by-step explanation:
f(x) = –3(25 – x²)/500 for –10 < x < +10
f(x) = (-75 + 3x²)/500 for -10 < x < 10
f(x) = (0.006x² - 0.15) for -10 < x < 10
a) The mean of a Probability distribution is given by its expected value
Mean = E(X) = Σ xᵢpᵢ
xᵢ = each variable In the sample space
pᵢ = probability of each variable In the sample space.
In integral terms,
E(X) = ∫ xf(x) dx (evaluating the integral all over the sample space; that is, for - 10 to 10)
E(X) = ∫¹⁰₋₁₀ xf(x) dx
E(X) = ∫¹⁰₋₁₀ x[0.006x² - 0.15] dx
E(X) = ∫¹⁰₋₁₀ [0.006x³ - 0.15x] dx
E(X) = [0.0015x⁴ - 0.075x²]¹⁰₋₁₀
E(X) = [0.0015(10⁴) - 0.075(10²)] - [0.0015(-10⁴) - 0.075(-10²)]
E(X) = [15 - 7.5] - [15 - 7.5] = 0 minutes
b) Variance is given as
Variance = Var(X) = Σx²p − μ²
where μ = mean = E(X)
In integral terms,
Var(X) = [∫¹⁰₋₁₀ x²f(x) dx] - μ²
∫¹⁰₋₁₀ x²f(x) dx = ∫¹⁰₋₁₀ x²(0.006x² - 0.15) dx
= ∫¹⁰₋₁₀ (0.006x⁴ - 0.15x²) dx
= [0.0012x⁵ - 0.05x³]¹⁰₋₁₀
= [0.0012(10⁵) - 0.05(10³)] - [0.0012(-10⁵) - 0.05(-10³)]
= [120 - 50] - [-120 + 50]
= 70 + 70 = 140
Var(X) = [∫¹⁰₋₁₀ x²f(x) dx] - μ²
Var(X) = 140 - 0² = 140.
Hope this Helps!!!
