Show that if X and Y are independent rv's, then E(XY) = E(X) · E(Y). [Hint: Consider the continuous case with f(x, y) = fX(x) · fY(y).] using sum x sum y xy * p(x,y) A surveyor wishes to lay out a square region with each side having length L. However, because of measurement error, he instead lays out a rectangle in which the north-south sides both have length X and the east-west sides both have length Y. Suppose that X and Y are independent and that each is uniformly distributed on the interval [L − A,L + A] (where 0 < A < L). What is the expected area of the resulting rectangle? E(area) =
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!
Show that if X and Y are independent rv's, then E(XY) = E(X) · E(Y). [Hint: Consider the continuous case with f(x, y) = fX(x) · fY(y).]
using sum x sum y xy * p(x,y)
A surveyor wishes to lay out a square region with each side having length L. However, because of measurement error, he instead lays out a rectangle in which the north-south sides both have length X and the east-west sides both have length Y. Suppose that X and Y are independent and that each is uniformly distributed on the interval [L − A,L + A] (where 0 < A < L). What is the expected area of the resulting rectangle?
E(area) =
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