Pearson eText for Probability & Statistics for Engineers and Scientists with R -- Instant Access (Pearson+)
1st Edition
ISBN: 9780137548552
Author: Michael Akritas
Publisher: PEARSON+
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6.54 Let Y₁, Y2,..., Y, be independent Poisson random variables with means 1, 2,..., An
respectively. Find the
a probability function of
Y.
b conditional probability function of Y₁, given that Y = m.
Y₁ = m.
c conditional probability function of Y₁+Y2, given that
6.55 Customers arrive at a department store checkout counter according to a Poisson distribution
with a mean of 7 per hour. In a given two-hour period, what is the probability that 20 or more
customers will arrive at the counter?
6.56 The length of time necessary to tune up a car is exponentially distributed with a mean of
.5 hour. If two cars are waiting for a tune-up and the service times are independent, what is
the probability that the total time for the two tune-ups will exceed 1.5 hours? [Hint: Recall the
result of Example 6.12.]
6.57 Let Y, Y2,..., Y,, be independent random variables such that each Y, has a gamma distribution
with parameters a, and B. That is, the distributions of the Y's might have different a's, but…
6.82
6.83
6.84
6.85
*6.86
6.87
If Y is a continuous random variable and m is the median of the distribution, then m is such
that P(Ym) = P(Y ≥ m) = 1/2. If Y₁, Y2,..., Y, are independent, exponentially dis-
tributed random variables with mean ẞ and median m, Example 6.17 implies that Y(n) =
max(Y₁, Y., Y) does not have an exponential distribution. Use the general form of FY() (y)
to show that P(Y(n) > m) = 1 - (.5)".
Refer to Exercise 6.82. If Y₁, Y2,..., Y,, is a random sample from any continuous distribution
with mean m, what is P(Y(n) > m)?
Refer to Exercise 6.26. The Weibull density function is given by
-my" m-le-y/a
f(y)= α
0.
y > 0,
elsewhere,
where a and m are positive constants. If a random sample of size n is taken from a Weibull
distributed population, find the distribution function and density function for Y(1) = min(Y1,
Y2,Y). Does Y(1) = have a Weibull distribution?
Let Y₁ and Y2 be independent and uniformly distributed over the interval (0, 1). Find
P(2Y(1) 0,
elsewhere,…
6.26
The Weibull density function is given by
e-y/a
f(y) = α
0.
y > 0,
elsewhere,
where a and m are positive constants. This density function is often used as a model for the
lengths of life of physical systems. Suppose Y has the Weibull density just given. Find
a the density function of UY".
b E(Y) for any positive integer k.
6.27
Let Y have an exponential distribution with mean ẞ.
6.28
6.29
a Prove that W = √Y has a Weibull density with α = ẞ and m = 2.
b
Use the result in Exercise 6.26(b) to give E(Yk/2) for any positive integer k.
Let Y have a uniform (0, 1) distribution. Show that U = -2ln(Y) has an exponential distri-
bution with mean 2.
The speed of a molecule in a uniform gas at equilibrium is a random variable V whose density
function is given by
6.30
6.31
6.32
f(v) = av²e-by², v > 0,
where b = m/2kT and k, T, and m denote Boltzmann's constant, the absolute temperature,
and the mass of the molecule, respectively.
a Derive the distribution of W = mV2/2, the kinetic energy of…
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