A health-food store stocks two different brands of a certain type of grain. Let X = theamount (lb) of brand A on hand and Y = the amount of brand B on hand. Suppose thejoint pdf of X and Y isf(x, y) =kxy x ≥ 0, y ≥ 0, 20 ≤ x + y ≤ 300 otherwisea. Draw the region of positive density and determine the value of k.b. Are X and Y independent? Answer by first deriving the marginal pdf of eachvariable.c. Compute P(X + Y ≤ 25).d. What is the expected total amount of this grain on hand?e. Compute Cov(X, Y) and Corr(X, Y).f. What is the variance of the total amount of grain on hand?
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!
A health-food store stocks two different brands of a certain type of grain. Let X = the
amount (lb) of brand A on hand and Y = the amount of brand B on hand. Suppose the
joint
f(x, y) =
kxy x ≥ 0, y ≥ 0, 20 ≤ x + y ≤ 30
0 otherwise
a. Draw the region of positive density and determine the value of k.
b. Are X and Y independent? Answer by first deriving the marginal pdf of each
variable.
c. Compute P(X + Y ≤ 25).
d. What is the expected total amount of this grain on hand?
e. Compute Cov(X, Y) and Corr(X, Y).
f. What is the variance of the total amount of grain on hand?
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