Joint probability density function is: x + y where 0<x<1 and 0<y<1 (0 elsewhere)
Find variance of (30x + 25y)
Answer is 11.48
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My calculations
Marginal probability density functions (same for x and y):
f(x) = x + 1/2 over 0<x<1
f(y) = y + 1/2 over 0<y<1
E(x) = E(y) = 7/12
E(x^2) = E(y^2) = 5/12
Var(x) = Var(y) = 11/144
E(x*y) = 1/3
Cov(x,y) = E(x*y) - E(x)*E(y) = -1/144
Var(30x + 25y) = 30^2 * Var(x) + 25^2 * Var(y) + 2 * 30 * 25 * Cov(x,y)
= 30^2 * 11/144 + 25^2 * 11/144 - 2 * 30 * 25 * 1/144
= 106.08
Where did I go wrong? Thanks!
To find the variance of the random variable Z = 30x + 25y, you correctly calculated the marginal probability density functions for x and y, as well as their expected values and variances. Let's review your calculations:
1. Marginal probability density functions:
f(x) = x + 1/2 for 0 < x < 1
f(y) = y + 1/2 for 0 < y < 1
2. Expected values:
E(x) = E(y) = ∫(x * f(x)) dx = ∫[(x * (x + 1/2))] dx = ∫(x^2 + x/2) dx = [x^3/3 + x^2/4] from 0 to 1 = (1/3 + 1/4) - (0/3 + 0/4) = 7/12
3. Expected squared values:
E(x^2) = E(y^2) = ∫(x^2 * f(x)) dx = ∫[(x^2 * (x + 1/2))] dx = ∫(x^3 + x^2/2) dx = [x^4/4 + x^3/6] from 0 to 1 = (1/4 + 1/6) - (0/4 + 0/6) = 5/12
4. Variances:
Var(x) = E(x^2) - [E(x)]^2 = 5/12 - (7/12)^2 = 5/12 - 49/144 = 11/144
Var(y) = E(y^2) - [E(y)]^2 = 5/12 - (7/12)^2 = 5/12 - 49/144 = 11/144
So far, everything is correct. However, where you made a mistake is when calculating the covariance between x and y:
5. Covariance:
Cov(x, y) = E(x * y) - E(x) * E(y) = ∫∫(x * y * f(x, y)) dx dy - E(x) * E(y)
Since the joint probability density function f(x, y) = x + y, for 0 < x < 1 and 0 < y < 1:
Cov(x, y) = ∫∫(x * y * (x + y)) dx dy - (7/12) * (7/12)
= ∫∫(x^2 * y + x * y^2) dx dy - 49/144
= [x^3 * y/3 + x^2 * y^2/2] from 0 to 1, 0 to 1 - 49/144
= (1/3 + 1/2) - (0/3 + 0/2) - 49/144
= 5/6 - 49/144
= 17/144
Now, let's recalculate the variance of Z = 30x + 25y:
Var(30x + 25y) = (30^2 * Var(x)) + (25^2 * Var(y)) + 2 * 30 * 25 * Cov(x, y)
= (900 * 11/144) + (625 * 11/144) + 2 * 30 * 25 * (17/144)
= (9900/144) + (6875/144) + (1020/12)
= 10450/144 + 6875/144 + 85/12
≈ 72.4306 + 47.7431 + 7.0833
≈ 127.257
Therefore, the correct variance of (30x + 25y) is approximately 127.257, not 11.48. It seems there was a calculation error in the step where you calculated the covariance between x and y.