The time between successive arrivals of buses may be modelled by a distribution whose density function is f(x) = 20² (x+0)³' x > 0. * If I arrive at the bus stop just as one bus is leaving, what is the probability that I have to wait at least x minutes for the next one? * Suppose that I have already been waiting for c minutes, where c < x. Find the probability that I have to wait at least x minutes altogether.

Suppose that the unknown X is ≥ 2  and has the probability density function fX(x)=Ce^−x,x ≥ 2.What is the numerical value of C?

Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.

The "kernel trick" is a quick way to integrate when you can recognize a distribution in some equation g(x) which you want to integrate. It takes advantage of the fact that the pdf f(x) of a proper probability distribution must integrate to 1 over the support. Thus if we can manipulate an equation g(x) into the pdf of a known distribution and some multiplying constant c in other words, g(x) = c · f(x) --- then we know that C • --- Saex 9(x)dx = c Smex f(x)dx = c ·1 = c Steps: 1. manipulate equation so that you can recognize the kernel of a distribution (the terms involving x); 2. use the distribution to figure out the normalizing constant; 3. multiply and divide by the distribution's normalizing constant, and rearrange so that inside the integral is the pdf of the new distribution, and outside are the constant terms 4. integrate over the support. The following questions will be much easier if you use the kernel trick, so this question is intended to give you basic practice. QUESTION:…