Show that the chain of independent random variables has a unique stationary distribution and find what it is.

A continuous random variable has a uniform distribution if its values are spread evenly over the range of possibilities.     true or false

What is the distribution of X + Y when X and Y are independent random variables with uniform distribution in (0, 1)?

Define Probabilities and Moments of Continuous Random Variables?

How shall we represent a mode for the continuous random variable?

Prove the theorem

If n independent random variables have the samegamma distribution with the parameters α and β, find themoment-generating function of their sum and, if possible,identify its distribution.

CHAPTER 5 TEXTBOOK Section 5.18- Exercise 20 Let X be the number of bacterial colonies per cubic centimeter, a Poisson random variable with expected value 3. (i) What is the probability that there is at least one bacterial colony in a randomly chosen cubic centimeter? (ii) What is the probability that in five randomly chosen cubic centimeters there is at least one cubic centimeter where there is at least one bacterial colony? (iii) How many cubic centimeters must be observed for the probability of observing at least one with at least one bacterial colony to be 0.95? 0.9999997 ÷ What is the probability that there is at least one bacterial colony in a randomly chosen cubic centimeter? What is the probability that in five randomly chosen cubic centimeters there is at least one cubic centimeter where 0.9502 there is at least one bacterial colony? How many cubic centimeters must be observed for the probability of observing at least one with at least one 1 bacterial colony to be 0.95?