Let Y be a random variable with the pdf 1 f(y) (y/3) exp(-y/6), 0
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!
![Let Y be a random variable with the pdf
1
(y/3) exp(-y/6), 0<y<∞,
ЗГ(2)22
0,
f (y)
elsewhere.
(a) Let X = Y/3. Find the pdf of X, name its distribution, and specify the associated
parameter value(s).
(b) Express X as a specific function of independent x2(1) random variables.
(c) Suppose that Z 4 N(0, 1), determine a specific function of Z that follows the x²(1)
distribution. Prove that such a function of Z follows a x²(1) distribution. [Hint:
T(1/2) = VT. You may use the distribution technique to find the CDF of the function
of Z. To derive the integration involved, you may need to change variable and note that
exp(-x2/2) is an even function of x.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2c0245b1-7c6d-4e39-b843-9dc1574eff46%2Fff0d588e-8d1d-4ca5-8d3f-913c36ebdd17%2Fd5vtetd_processed.jpeg&w=3840&q=75)
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