Answer:
a) 0.0625 = 6.25%
b) 106.67 Ω
c) 9.43 Ω
d) 1
Explanation:
The probability distribution is given as
f(x) = (x - 80)/800 for 80 < x < 120
f(x) = 0 otherwise.
f(x) = (x/800) - (0.1)
a) Proportion of resistors with resistance less than 90 Ω
P(X < 90) = ∫⁹⁰₈₀ f(x) dx
∫⁹⁰₈₀ f(x) dx = ∫⁹⁰₈₀ [(x/800) - (0.1)]
= [(x²/1600) - 0.1x]⁹⁰₈₀
= [(90²/1600) - 0.1(90)] - [(80²/1600) - 0.1(80)]
= (5.0625 - 9) - [4 - 8]
= -3.9375 + 4 = 0.0625 = 6.25%
b) The mean is given by the expected value expression E(X) = = Σ xᵢpᵢ (with the sum done all over the data set for each variable and its corresponding probability)
It can be written in integral form as
Mean = ∫¹²⁰₈₀ xf(x) dx (with the integral done all over the probability function, i.e. from, 80 to 120)
Mean = ∫¹²⁰₈₀ x[(x/800) - (0.1)] dx
= ∫¹²⁰₈₀ [(x²/800) - (0.1x)] dx
= [(x³/2400) - (0.05x²)]¹²⁰₈₀
= [(120³/2400) - (0.05(120²)] - [(80³/2400) - (0.05(80²)]
= [720 - 720] - [213.33 - 320] = 106.67 Ω
c) Standard deviation = √(variance)
Variance = Var(X) = Σx²p − μ²
μ = mean = expected value = 106.67 Ω
Σx²p = ∫¹²⁰₈₀ x²f(x) dx = ∫¹²⁰₈₀ x² [(x/800) - (0.1)] dx = ∫¹²⁰₈₀ [(x³/800) - (0.1x²)] dx
= [(x⁴/3200) - (0.0333x³)]¹²⁰₈₀
= [(120⁴/3200) - (0.0333(120³)] - [(80⁴/3200) - (0.0333(80)³)]
= (64800 - 57600) - (12800 - 17066.667)
= 11466.667
Variance = 11466.667 - 106.67² = 88.85
Standard deviation = √88.85 = 9.43 Ω
d) Cdf = sum of probabilities over the entire probability function
Cdf = ∫¹²⁰₈₀ f(x) dx = ∫¹²⁰₈₀ [(x/800) - (0.1)] dx
= [(x²/1600) - 0.1x]¹²⁰₈₀ = [(120²/1600) - 0.1(120)] - [(80²/1600) - 0.1(80)] = (9 - 12) - (4 - 8) = -3+4 = 1 as it should be!!!
Hope this Helps!!!
