Suppose that the c.d.f. of a random variable X is F(x) = 1-e^(-sqrt(x)), x>=0. a) Find the probability density function of X. b) Compute the quantile function of X. c) How can X be generated from a uniform random number generator?

a) The probability density function (pdf) of a random variable X can be found by taking the derivative of its cumulative distribution function (c.d.f.). In this case, the c.d.f. is F(x) = 1 - e^(-sqrt(x)), for x >= 0.

The derivative of F(x) with respect to x is f(x) = F'(x) = 0.5 * e^(-sqrt(x)) / sqrt(x), for x > 0, and 0 for x = 0.

b) The quantile function, also known as the inverse cumulative distribution function, is the function Q(p) that gives the value x such that F(x) = p. To find Q(p), we set F(x) = p and solve for x:

1 - e^(-sqrt(x)) = p e^(-sqrt(x)) = 1 - p -sqrt(x) = ln(1 - p) x = (-ln(1 - p))^2

So, the quantile function is Q(p) = (-ln(1 - p))^2.

c) To generate a random variable X from a uniform random number generator, we can use the inverse transform method. This method involves generating a random number U from a uniform distribution on the interval (0, 1), and then transforming U using the quantile function of X to get a random number with the same distribution as X.

In this case, if U is a random number from a uniform distribution on (0, 1), then X = Q(U) = (-ln(1 - U))^2 is a random number with the same distribution as X.