Let X be a random variable with density functionfX (x) = k1 + x2 , −∞ < x < ∞.Determine k and the distribution function

To find the value of k such that the function f is a probability density function, we need to ensure that the integral of f from -∞ to ∞ is equal to 1, because the total probability must be 1.

The function f is non-zero for all real numbers, so we need to integrate from -∞ to ∞:

∫ from -∞ to ∞ of k/(1 + x^2) dx.

This is a standard integral that evaluates to πk. Setting this equal to 1 gives πk = 1, so k = 1/π.

The distribution function, also known as the cumulative distribution function (CDF), is found by integrating the density function from -∞ to x:

F(x) = ∫ from -∞ to x of f(t) dt = ∫ from -∞ to x of (1/π)/(1 + t^2) dt.

This is also a standard integral, and it evaluates to F(x) = (1/2) + (1/π)arctan(x) for -∞ < x < ∞.