The operator of a pumping station has observed that demand for water during early afternoon hours has an approximately exponential distribution with mean 1000 cfs (cubic feet per second). a) Find the probability that the demand will exceed 700 cfs during the early afternoon on a randomly selected day. b) What water-pumping capacity should the station maintain during early afternoons so that the probability that demand will be below the capacity on a randomly selected day is 0.995? c) Of the three randomly selected afternoons, what is the probability that on at least two afternoons the demand will exceed 700 cfs?
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!
The operator of a pumping station has observed that demand for water
during early afternoon hours has an approximately exponential distribution with mean 1000
cfs (cubic feet per second).
a) Find the probability that the demand will exceed 700 cfs during the early afternoon
on a randomly selected day.
b) What water-pumping capacity should the station maintain during early afternoons so
that the probability that demand will be below the capacity on a randomly selected
day is 0.995?
c) Of the three randomly selected afternoons, what is the probability that on at least two
afternoons the demand will exceed 700 cfs?
2. Let Y1 and Y2 be random variables with joint density function
f(y1, y2) = (6/7(y^2+y1y2/2) 0 < y1 < 1, 0 < y2 < 2,
0, elsewhere
a) Find marginal density
b) Find P(0 < Y1 < 0.3, −2 < Y2 < 1).
c) Find P(0.6 < Y1 < 1|0 < Y2 < 1).
3.The joint density function of Y1 and Y2 is given by
f(y1, y2) = (y1 + y2), 0 < y1 < 1, 0 < y2 < 1,
0, elsewhere
a) Find marginal density functions. Are Y1 and Y2 independent?
b) Could you tell whether Y1 and Y2 were independent or not without finding marginal
density functions? Explain.
c) Find P(Y1 + Y2 < 0.2).
d) Find P(0 < Y2 < 0.5|Y1 = 0.4).
4.The joint probability distribution of Y1 and Y2 is given by
f(y1, y2) = (6*y1^2*y2, 0 ≤ y1 ≤ y2, y1 + y2 ≤ 2
0, elsewhere
a) Find marginal density functions. Are Y1 and Y2 independent?
b) Find P(0 < Y1 < 1, 1 < Y2 < 2).
c) Find P(Y2 > Y1).
Suppose we have the following joint density function
f(y1, y2) = (6(1 − y2), 0 ≤ y1 ≤ y2 < 1,
0, elsewhere
a) Find marginal density functions.
b) Find P(0.2 ≤ Y1 ≤ 0.5, 0 < Y2 < 1)
c) Find P(Y1 > 0, Y2 < 0.2)
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