I1. Suppose that a random sample of sizen is to be taken from the normal distribution with mean a and standard deviation 2. Determine the smallest value of n such that Pr, -l <0.1)20.9.
I1. Suppose that a random sample of sizen is to be taken from the normal distribution with mean a and standard deviation 2. Determine the smallest value of n such that Pr, -l <0.1)20.9.
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Concept explainers
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
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question 11
Transcribed Image Text:The p.d..s of the lognormal distributions will be Tound in Excrcise 17 of this
section. The c.d.f. of each lognormal distribution is easily constructed from the
standard normal c.d.f. 4. Let X have the lognormal distribution with parameters
u and o. Then
Pr(X sx) = Prlog(X) s logéx)) = a (logts) –
The results from earlier in this section about linear combinations of normal random
variables translate into results about products of powers of lognormal random vari-
ables Results about sums of independent normal random variables translate into
results about products of independent lognormal random variables
Summary
We introduced the family of normal distributions. The parameters of each normal
distribution are its mean and variance. A linear combination of independent normal
random variables has the normal distribution with mean equal to the linear combi-
nation of the means and variance determined by Corollary 4.3.1. In particular, if X
has the normal distribution with mean u and variance a?, then (X -)/a has the
standard normal distribution (mean 0 and variance 1). Probabilities and quantiles for
normal distributions can be obtained from tables or computer programs for standard
normal probabilities and quantiles. For example, if X has the normal distribution with
mean a and variance o, then the c.d.f. of X is F(x) = (x - /a) and the quantile
function of X is F-p) = +-a, where o is the standard normal c.d.f.
56 The Normal Distributions 315
Exercises
1. Find the 0.5, 0.25, 0.75, 0.1, and 0.9 quantiles of the
standard normal distribution.
2. Suppose that X has the normal distribution for which
the mean is 1 and the variance is 4. Find the value of each
of the following probabilities:
a. Pr(X 53)
e. Pr(X- 1)
e. Pr(X 2 0)
10. If a random sample of 25 observations is taken from
the normal distribution with nmean a and standard devia-
tion 2, what is the probability that the sample mean will
lie within one unit of a?
11. Suppose that a random sample of size n is to be taken
from the normal distribution with mean a and standard
deviation 2. Determine the smallest value of a such that
b. Pr(X > 1.5)
d. Pr(2< X <S)
PrX, - al <0.1) 2 0.9.
I. Pr(-1< X < 0.5)
. PrX| 5 2) h. Prds-2x +35R)
12.
3. If the temperature in degrees Fahrenheit at a certain
location is normally distributed with a mean of 68 degrees
and a standard deviation of 4 degrees, what is the distri-
bution of the temperature in degrees Celsius at the same
location?
a. Sketch the cd.f. of the standard normal distribu-
tion from the values given in the table at the end of
this book.
h. From the sketch given in part (a) of this exercise,
sketch the c.d.f. of the normal distribution for which
the mean is-2 and the standard deviation is 3.
4. Find the 0.25 and 0.75 quantiles of the Fahrenheit tem-
perature at the location mentioned in Exercise 3.
5. Let X,. X2, and X, be independent lifetimes of memory
chips. Suppose that each X, has the nomal distribution
with mean 300 hours and standard deviation 10 hours
13. Suppose that the diameters of the bolts in a large box
follow a normal distribution with a mean of 2 centimeters
and a standard deviation of 0.03 centimeter. Also, suppose
that the diameters of the holes in the nuts in another large
box follow the normal distribution with a mean of 2.02
Compute the probability that at least one of the three
chips lasts at least 290 hours
centimeters and a standard deviation of 0.04 centimeter.
6. If the m.g.f. of a random variable X is vi)- for
- I, what is the distribution of X?
7. Suppose that the measured voltage in a certain electric
A bolt and a nut will fit together if the diameter of the
hole in the nut is greater than the diameter of the bolt and
the difference between these diameters is not greater than
0.05 centimeter. If a bolt and a nut are selected at random.
circuit has the normal distribution with mean 120 and
what is the probability that they will fit together?
standard deviation 2. If three independent measurements
of the voltage are made, what is the probability that all
three measurements will lie between 116 and 118?
14. Suppose that on a certain examination in advanced
mathematics, students from university A achieve scores
that are normally distributed with a mean of 625 and a
variance of 100, and students from university 8 achieve
scores which are normally distributed with a mean of 600
and a variance of 150. If two students from university A
and three students from university 8 take this examina-
tion, what is the probability that the average of the scores
of the two students from university A will be greater than
the average of the scores of the three students from univer-
sity 8? Hint: Determine the distribution of the difference
between the two averages
8. Evaluate the integral dx.
9. A straight rod is formed by connecting three sections
A, B, and C, each of which is manufactured on a different
machine. The length of section A, in inches, has the normal
distribution with mean 20 and variance 0.04. The length of
section B, in inches, has the normal distribution with mean
14 and variance 0.01. The length of section C, in inches, has
the normal distribution with mean 26 and variance 0.04.
As indicated in Fig. 5.6, the three sections are joined so
that there is an overlap of 2 inches at each connection.
Suppose that the rod can be used in the construction of an
55.7
15. Suppose that 10 percent of the people in a certain
population have the eye disease glaucoma. For persoons
who have glaucoma, measurements of eye pressure X will
be normally distributed with a mean of 25 and a variance
of 1. For persons who do not have glaucoma, the pressure
X will be normally distributed with a mean of 20 and a
variance of 1. Suppose that a person is selected at random
from the population and her eye presure X is measured.
a. Determine the conditional probability that the per
son has glaucoma given that X
h. For what values of x is the conditional probability in
part (a) greater than 1/2?
airplane wing if its total length in inches is between
and S6.3. What is the probability that the rod can be used?
Figure 5.6 Sections of the rod in Exercise 9.
316
Chapter S Special Distributions
16. Suppose that the joint p.d.f. of two random variables
X and Y is
23. Suppose that x has the lognormal distribution with
parameters 4.1 and 8. Find the distribution of 3X2
24. The method of completing the square is used several
times in this text. It is a useful method for combining
several quadratic and linear polynomials into a perfect
square plus a constant. Prove the following identity, which
is one general form of completing the square:
f(x, y) e1/b4y) for - N
and - y< N.
Find Pr-Vi«X +Y < 2/2).
17. Consider a random variable X having the lognormal
distribution with parameters a and o. Determine the
p.d.f. of X.
18. Suppose that the random variables X and Y are inde-
pendent and that each has the standard normal distribu-
tion. Show that the quotient X/Y has the Cauchy distri-
bution.
-(E-)(-
19 Suppose that he measurenent Y of presure mude hw
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