You are given the probability distribution function (PDF) of a continuous random variable X is fX(x). Let Y be a continuous random variable such that Y = aX + b, where a and b are non-zero real constants. 1. Find the PDF of Y in terms of fX , a, and b. 2. Let X be an exponential random variable with parameter λ. When will Y also be an exponential random variable? 3. Let X be a normal random variable with mean μ and variance σ2 . When will Y also be a normal random variable?
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
You are given the
1. Find the PDF of Y in terms of fX , a, and b.
2. Let X be an exponential random variable with parameter λ. When will Y
also be an exponential random variable?
3. Let X be a normal random variable with mean μ and variance σ2 . When
will Y also be a normal random variable?
Trending now
This is a popular solution!
Step by step
Solved in 4 steps
