Question

for certain ore samples, the proportion y of impurities per sample is a random variable with density function w = 5-5.y.Find the mean and variance of w .

Answer 1

To find the mean and variance of the random variable w, use the given density function and calculate the expected value and variance based on it. The mean is 25/3 and the variance is 25/18.

Step-by-step explanation:

To find the mean and variance of the random variable w, we first need to find its probability density function (PDF). The given density function is w = 5 - 5y. To find the mean, we need to calculate the expected value of w, which is given by:

E(w) = ∫w * f(w) dw, where f(w) is the PDF of w.

Substituting the given density function, we have:

E(w) = ∫(5 - 5y)(5 - 5y) dy

After integrating and simplifying, we get:

E(w) = 25/3

To find the variance, we need to calculate E(w^2) - (E(w))^2. Substituting the given density function, we have:

E(w^2) = ∫(w^2)(5 - 5y) dy

After integrating and simplifying, we get:

E(w^2) = 125/6

Finally, the variance is:

Var(w) = E(w^2) - (E(w))^2 = 125/6 - (25/3)^2 = 25/18.