4.2P The effectiveness of solar-energy heating units depends on the amount of radiation available from the sun. For a typical October, daily total solar radiation in Tampa, Florida, approximately follows the probability density function given below (units are hundreds of calories): ;(x – 2)(6 – x) 2

A company uses wood chips in the manufacture of processed wood pieces of various sizes. Suppose the tons of wood chips T used per day has a probability density function f(T) = 0.25e-0.25T (a) Find the probability of using more than 4 tons of wood chips in a day. (Round your answer to three decimal places.) (b) Find the number of tons 7 (to the nearest ton) so that the probability of using more than 7 tons in a day is 0.07. tons

(i) Attempt the following questions. Show all necessary steps. Read the questions carefully. (ii) Any copied material from internet or your class fellows will get you ZERO marks. Q. 1 Let X be the number of bugs in the first draft of a piece of software, and let Y be the number of bugs remaining in the second draft of the same software. Their joint probability mass function is given by the following table: Y 0. 1 3 0. 0.03 0.14 0.09 0.08 1 0.05 0.10 K 0.04 0.04 0.09 0.08 0.09 (i) Obtain K so that it is the joint probability mass function; (ii) Find cumulative distribution function Fyy (x, y); (iii) Find the following probabilities using cdf obtained above (a) P(X 4), (c) P(X + Y > 3), (e) P(X < 3,Y < 2) 21

The diameter of a particle of contamination (in micrometers) is modeled with the probability density function 2 f(x)= = x3 for x > 1 . What is the probability that the diameter of the particle of contamination is below 2 micrometers?