Consider the rational number 0. 9 ¯ . a. Use the method from Example 8 on page 230 to convert 0. 9 ¯ to a quotient of integers. b. Find a number halfway between 0. 9 ¯ and 1 by adding the two numbers and dividing by 2. c. Find 1 3 + 2 3 by adding the fractions. Now express 1 3 and 2 3 as repeating decimals and find the sum using the decimal representation of 2 3 and 2 3 . d. What conclusion can you draw from parts (a), (b), and (c)?
Consider the rational number 0. 9 ¯ . a. Use the method from Example 8 on page 230 to convert 0. 9 ¯ to a quotient of integers. b. Find a number halfway between 0. 9 ¯ and 1 by adding the two numbers and dividing by 2. c. Find 1 3 + 2 3 by adding the fractions. Now express 1 3 and 2 3 as repeating decimals and find the sum using the decimal representation of 2 3 and 2 3 . d. What conclusion can you draw from parts (a), (b), and (c)?
Solution Summary: The author explains how the number 0.stackrel 9 is converted to a quotient of integers.
a. Use the method from Example 8 on page 230 to convert
0.
9
¯
to a quotient of integers.
b. Find a number halfway between
0.
9
¯
and 1 by adding the two numbers and dividing by 2.
c. Find
1
3
+
2
3
by adding the fractions. Now express
1
3
and
2
3
as repeating decimals and find the sum using the decimal representation of
2
3
and
2
3
.
d. What conclusion can you draw from parts (a), (b), and (c)?
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…
Please ensure that all parts of the question are answered thoroughly and clearly. Include a diagram to help explain answers. Make sure the explanation is easy to follow. Would appreciate work done written on paper. Thank you.
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,…
Chapter 5 Solutions
A Survey of Mathematics with Applications plus MyLab Math Student Access Card -- Access Code Card Package (10th Edition)
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