Question

2. The joint density function of X and Y is given by f(x,y) = { (1/2exp(-x))+(cy/(1+x)²)} range of X is 0<=x an Y is 0<=y<=1

a)Find constant c
B)find P(0<=x<=1,0<=y0<=1/2)​

Answer 1

The question involves finding the constant c in a joint probability density function and calculating a specific probability. This requires using the property that the integral of the pdf over all possible values must equal 1 and then integrating the pdf over the specified ranges to find the probability.

Step-by-step explanation:

The question asks to find the value of the constant c in the joint density function f(x,y), and to compute the probability P(0≤x≤1,0≤y≤1/2), concerning the random variables X and Y. We start by acknowledging that the joint probability density function (pdf) must integrate to 1 over the range of X and Y.

To find the constant c, the integral of f(x,y) with respect to y from 0 to 1, and then with respect to x from 0 to infinity, should equal 1. We solve for c using this formula:

1 = ∫∫ f(x,y) dydx

This will yield the value of the constant c that normalizes the pdf.

To calculate P(0≤x≤1,0≤y≤1/2), we integrate the pdf over the given range for x and y:

P(0≤x≤1,0≤y≤1/2) = ∫∫ f(x,y) dydx

If X had an exponential distribution, the cumulative distribution function would be represented as P(X < x) = 1 – e(−0.5)(x), for example. However, we deal with a joint distribution here, so the calculation will involve both X and Y variables.

Once we determine c, we use the joint pdf to compute the desired probability by setting up and evaluating the double integral over the given bounds for X and Y.