The “random” parts of the algorithm in Self-Test Problem 6.9 &1 can be written in terms of the generated values of a sequence of independent uniform (0, 1) random variables, known as random numbers. With [x] defined as the largest integer less than or equal to x, the first step can be written as follows: Step 1. Generate a uniform (0, 1) random variable U. Let X = [ m U ] + 1 and determine the value of n ( X ) . a. Explain why the above is equivalent to step I of Problem 6.8. Hint: What is the probability mass function of X? b. Write the remaining steps of the algorithm in a similar style.
The “random” parts of the algorithm in Self-Test Problem 6.9 &1 can be written in terms of the generated values of a sequence of independent uniform (0, 1) random variables, known as random numbers. With [x] defined as the largest integer less than or equal to x, the first step can be written as follows: Step 1. Generate a uniform (0, 1) random variable U. Let X = [ m U ] + 1 and determine the value of n ( X ) . a. Explain why the above is equivalent to step I of Problem 6.8. Hint: What is the probability mass function of X? b. Write the remaining steps of the algorithm in a similar style.
Solution Summary: The author explains the relation between probability mass function and probability of selecting page out of m pages.
The “random” parts of the algorithm in Self-Test Problem 6.9 &1 can be written in terms of the generated values of a sequence of independent uniform (0, 1) random variables, known as random numbers. With [x] defined as the largest integer less than or equal to x, the first step can be written as follows:
Step 1. Generate a uniform (0, 1) random variable U. Let
X
=
[
m
U
]
+
1
and determine the value of
n
(
X
)
.
a. Explain why the above is equivalent to step I of Problem 6.8.
Hint: What is the probability mass function of X?
b. Write the remaining steps of the algorithm in a similar style.
Expression, rule, or law that gives the relationship between an independent variable and dependent variable. Some important types of functions are injective function, surjective function, polynomial function, and inverse function.
A certain type of digital camera comes in either a 3-megapixel version or a 4-megapixel version. A camera store has received a shipment of 15 of these cameras, of which 6 have a 3-megapixel resolution. Suppose that 5 of these cameras are randomly selected to be stored behind the counter; the other 10 are placed in a storeroom. Let X= the number of 3-megapixel cameras among the 5 selected for behind-the-counter storage.
a. What kind of a distribution does X have (name and values of all parameters)?
b. Compute P(X=2), P(X≤2), and P(X≥2).
c. Calculate the mean value and standard deviation of X.
Each of 14 refrigerators of a certain type has been returned to a distributor because of an audible, high-pitched, oscillating noise when the refrigerators are running. Suppose that 9 of these refrigerators have a defective compressor and the other 5 have less serious problems. If the refrigerators are examined in random order, let X be the number among the first 6 examined that have a defective compressor.
(I have figured out part "a" but need help with "b" and P(X ≤ 3) in "c")
(a)
Calculate
P(X = 4) and P(X ≤ 4).
(Round your answers to four decimal places.)
P(X = 4)
=
P(X ≤ 4)
=
(b)
Determine the probability that X exceeds its mean value by more than 1 standard deviation. (Round your answer to four decimal places.)
(c)
Consider a large shipment of 400 refrigerators, of which 40 have defective compressors. If X is the number among 25 randomly selected refrigerators that have defective compressors, describe a less tedious way to calculate (at least approximately)…
A certain type of digital camera comes in either a 3-megapixel version or a 4-megapixel version. A camera store has received a shipment of 14 of these cameras, of which 6 have 3-megapixel resolution. Suppose that 3 of these cameras are randomly selected to be
stored behind the counter; the other 11 are placed in a storeroom. Let X = the number of 3-megapixel cameras among the 3 selected for behind-the-counter storage.
(a) What kind of distribution does X have (name and values of all parameters)?
Distribution
Parameters
O binomial
O n = 3
O geometric
O p = 3/14
hypergeometric
O N = 14
O negative binomial
O M = 6
O µ = 9/14
O 2 = 9/28
(b) Compute the following. (Enter your answers as fractions.)
P(X = 2)
P(X S 2)
P(X 2 2)
(c) Calculate the mean value and standard deviation of X. (Give your answers to three decimal places.)
mean value
standard deviation
O O O
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