Question

Suppose that the random variables X and Y have the following joint probability density function: f(x,y)=ce ⁻⁴ˣ⁻³ʸ, where 0

Find the value of c.

Answer 1

To find the value of c for the given joint pdf, integrate f(x, y) = ce⁻⁴x⁻⁵y over the domain of X and Y and set this integral to 1, then solve for c.

Step-by-step explanation:

The student asked to find the value of c for the joint probability density function (pdf) f(x,y) = ce⁻⁴x⁻⁵y, defined for the random variables X and Y and with the condition 0 < x < ∞, 0 < y < ∞. To determine the value of c, we must ensure that the total probability integrates to 1 over the domain of X and Y. This involves setting up an integral over the range of both variables and solving for c:

Calculate the double integral of f(x, y) over the relevant domain.

Set this integral equal to 1 (since the total probability must be 1).

Solve the resulting equation for the constant c.

The answer involves calculus, specifically the use of double integration to evaluate the area under a two-variable probability density function.

Answer 2

To determine the value of 'c' for the joint pdf, we integrate the function over all possible values of 'X' and 'Y', and set the integral equal to 1 to satisfy the normalization condition of probability density functions. The double integration over the domains of 'X' and 'Y' results in an equation in terms of 'c', which can be solved to ensure the total probability is 1.

Step-by-step explanation:

The question asks to determine the value of c for the given joint probability density function (pdf) f(x, y) = ce−4x−3y, where 0 < x < ∞ and 0 < y < ∞. To find c, we must use the fact that the integral of the pdf over the entire range of X and Y must equal 1, since the total probability must sum to 1. This is an application of the normalization condition for probability density functions. The continuous random variables undergo integration across their respective domains, which, in this case, are both from 0 to ∞. This process involves setting up a double integral of the pdf f(x, y) with respect to x and y and solving for c, making sure the result of the integral equals 1. The steps to solve this would include:

Setting up the double integral of the function f(x, y) over their respective ranges.

Applying integration techniques to evaluate the integral.

Solving the resulting equation for c so that the total integral equals 1, which is the requirement for a probability density function.

Therefore, once the double integral is computed, you obtain an equation in terms of c and solve for c to ensure that the joint pdf is properly normalized.