Determine the value of c that makes the function f(x, y) = c(x + y) a joint probability density function over the range 0 Determine the following: (a) P(X1,Y2) (b) P(1 < X &l…

Question

Determine the value of c that makes the function f(x, y) = c(x + y) a joint probability density function over the range 0

Determine the following:

(a) P(X1,Y2)

(b) P(1 < X < 2)

(c) P(Y> 1)

(d) P(X2,Y2)

(e) E(X)

(f) V(X)

(g) Marginal probability distribution of X

(h) Conditional probability distribution of Y given that X = 1

(i) E(Y|X = 1)

(j) P(Y2X = 1)

(k) Conditional probability distribution of X given that Y 52

Answer 1

To find the value of c for the joint probability density function, set up the double integral of c(x + y) over x and y from 0 to 1 and solve for c such that the total area under the curve is 1. Subsequently, this function can be used for various probability and expectation calculations.

Step-by-step explanation:

The student is asking to find the value of c that makes the function f(x, y) = c(x + y) a joint probability density function (PDF) over the range 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1. To ensure f(x, y) is a PDF, the double integral of f(x, y) over the range of x and y must equal 1. By calculating the double integral, we find the constant c that satisfies this condition.

Let's perform the integration:

1. Set up the double integral ∫∫D f(x, y) dx dy, where D is the domain 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.
2. Integrate f(x, y) = c(x + y) first with respect to x from 0 to 1, then with respect to y from 0 to 1.
3. Solve for c to make the total area (total probability) equal to 1.

Note that after determining c, you can use this density function to calculate various probabilities and expectations such as P(X > 1, Y > 2), E(X), E(Y|X = 1), V(X), and marginal probabilities.