(Solved): a. The joint probability density function of the random variables X and Y is f(x,y)={k(1xy)0 ...

Expert Answer

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To find the value of k, we need to use the normalization condition that the integral of the common probability viscosity function over the entire range must equal 1. Integrating the given function f(x, y) over its valid range gives:



















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Thus, the value of k is 6.



1. The given question involves a common probability viscosity function of two arbitrary variables, X and Y
2. Integrating the given function f(x, y) over its valid range involves nested integrals. We first integrate with respect to y from 0 to 1-x, and then with respect to x from 0 to 1.
3. Assessing the inner integral with respect to y gives( y-(1/2) y2) estimated at the limits 0 and 1-x.
4. Simplifying the inner integral yields (1-x) - (1/2)(1-x)^2.
5. Integrating the simplified inner integral with respect to x gives [x - (1/2)x^2 - (1/2)(1/3)(1-x)^3] evaluated at the limits 0 and 1.
6. Simplifying the outer integral gives (1 - 1/2 - (1/2)(1/3)).
7. Solving the equation k/6 = 1, we find that k = 6.
Therefore, the value of k is 6, satisfying the normalization condition.