Let the random variables and have joint pdf as follows:

Find (round off to third decimal place).

Question

Let the random variables  and  have joint pdf as follows:

Find  (round off to third decimal place).

BlackTom AI · Accepted Answer

We start by restating the given joint pdf and the domain for the variables: f(x,y) = (1/5) (11x^2 + 4y^2) for 0 < x < 1 and 0 < y < 1.
To find Cov(X,Y) = E[XY] − E[X]E[Y], we need E[X], E[Y], and E[XY].
First, compute E[X]: E[X] = ∫_{0}^{1} ∫_{0}^{1} x · f(x,y) dy dx. Inside the integral over y, f becomes (1/5) x (11x^2 + 4y^2). Integrating with respect to y from 0 to 1 gives (1/5) x [11x^2 · 1 + 4 · ∫_{0}^{1} y^2 dy] = (1/5) x [11x^2 + 4/3].
Now integrate with respect to x: ∫_{0}^{1} (1/5) x [11x^2 + 4/3] dx = (1/5) [11 ∫_{0}^{1} x^3 dx + (4/3) ∫_{0}^{1} x dx] = (1/5) [11(1/4) + (4/3)(1/2)] = (1/5) [11/4 + 2/3] = (1/5) · (41/12) = 41/60.
Next, compute E[Y]: E[Y] = ∫_{0}^{1} ∫_{0}^{1} y · f(x,y) dx dy. Inner integral over x gives (1/5) y [11 ∫_{0}^{1} x^2 dx + 4y^2 ∫_{0}^{1} dx] = (1/5) y [11/3 + 4y^2]. Then integrate over y: ∫_{0}^{1} (1/5) y [11/3 + 4y^2] dy = (1/5) [ (11/3) ∫_{0}^{1} y dy + 4 ∫_{0}^{1} y^3 dy ] = (1/5) [ (11/3)(1/2) + 4(1/4) ] = (1/5) [11/6 + 1] = (1/5) · (17/6) = 17/30.
Now determine E[XY]: E[XY] = ∫_{0}^{1} ∫_{0}^{1} x y · f(x,y) dx dy. Inner integral over x: (1/5) y ∫_{0}^{1} x (11x^2 + 4y^2) dx = (1/5) y [11 ∫_{0}^{1} x^3 dx + 4y^2 ∫_{0}^{1} x dx] = (1/5) y [11(1/4) + 4y^2(1/2)] = (1/5) y [11/4 + 2y^2]. Then integrate over y: ∫_{0}^{1} (1/5) y [11/4 + 2y^2] dy = (1/5) [ (11/4) ∫_{0}^{1} y dy + 2 ∫_{0}^{1} y^3 dy ] = (1/5) [ (11/4)(1/2) + 2(1/4) ] = (1/5) [11/8 + 1/2] = (1/5) · (15/8) = 3/8.
With these pieces, Cov(X,Y) = E[XY] − E[X]E[Y] = (3/8) − (41/60)(17/30).
Compute the product: (41/60)(17/30) = 697/1800 ≈ 0.387222..., and 3/8 = 0.375. Therefore Cov(X,Y) = 0.375 − 697/1800 = (675/1800 − 697/1800) = −22/1800 = −11/900 ≈ −0.0122. Rounding to three decimal places yields −0.012.

AMS 310 Homework 7

Let the random variables and have joint pdf as follows: Find (round off to third decimal place).

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