Introduction to Probability and Statistics

William Mendenhall, III, Robert J. Beaver, Barbara M. Beaver

Chapter 6

The Normal Probability Distribution - all with Video Answers

Educators

Section 1

Probability Distributions for Continuous Random Variables

Problem 1

Let $x$ have a uniform distribution on the interval 0 to $10 .$ Find the probabilities.
$$
P(x<5)
$$

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Problem 2

Let $x$ have a uniform distribution on the interval 0 to $10 .$ Find the probabilities.
$$
P(3<x<7)
$$

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Problem 3

Let $x$ have a uniform distribution on the interval 0 to $10 .$ Find the probabilities.
$$
P(x>8)
$$

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Problem 4

Let $x$ have a uniform distribution on the interval 0 to $10 .$ Find the probabilities.
$$
P(2.5<x<8.3)
$$

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Problem 5

Suppose $x$ has a uniform distribution on the interval from -1 to $1 .$ Find the probabilities.
$$
P(x<0)
$$

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Problem 6

Suppose $x$ has a uniform distribution on the interval from -1 to $1 .$ Find the probabilities.
$$
P(x>.7)
$$

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Problem 7

Suppose $x$ has a uniform distribution on the interval from -1 to $1 .$ Find the probabilities.
$$
P(-.5<x<.5)
$$

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Problem 8

Suppose $x$ has a uniform distribution on the interval from -1 to $1 .$ Find the probabilities.
$$
P(-.7<x<.2)
$$

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Problem 9

Let $x$ have an exponential distribution with $\lambda=1$. Find the probabilities.
$$
P(x>1)
$$

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Problem 10

Let $x$ have an exponential distribution with $\lambda=1$. Find the probabilities.
$$
P(1<x<5)
$$

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Problem 11

Let $x$ have an exponential distribution with $\lambda=1$. Find the probabilities.
$$
P(x<1.5)
$$

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Problem 12

Let $x$ have an exponential distribution with $\lambda=1$. Find the probabilities.
$$
P(2<x<4)
$$

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Problem 13

Let $x$ have an exponential distribution with $\lambda=0.2$. Find the probabilities.
$$
P(x>6)
$$

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Problem 14

Let $x$ have an exponential distribution with $\lambda=0.2$. Find the probabilities.
$$
P(4<x<6)
$$

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Problem 15

Let $x$ have an exponential distribution with $\lambda=0.2$. Find the probabilities.
$$
P(x<5)
$$

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Problem 16

Let $x$ have an exponential distribution with $\lambda=0.2$. Find the probabilities.
$$
P(3<x<8)
$$

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Problem 17

You arrive at a bus stop to wait for a bus that comes by once every 30 minutes. You don't know what time the last bus came by. The time $x$ that you wait before the bus arrives is uniformly distributed on the interval from 0 to 30 minutes.
a. What is the probability that you will have to wait longer than 20 minutes?
b. What is the probability that you will have to wait less than 10 minutes?
c. What is the probability that you will wait between 10 and 20 minutes?

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Problem 18

The thickness in microns $(\mu)$ of a protective coating applied to a conductor designed to work in corrosive conditions is uniformly distributed on the interval from 25 to $50 .$
a. What is the probability that the thickness of the coating is greater than 45 microns?
b. What is the probability that the thickness of the coating is between 35 and 45 microns?
c. What is the probability that the thickness of the coating is less than 40 microns?

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Problem 19

The length of life (in days) of an alkaline battery has an exponential distribution with an average life of 1 year, so that $\lambda=1 / 365$.
a. What is the probability that an alkaline battery will fail before 180 days?
b. What is the probability that an alkaline battery will last beyond 1 year?
c. If a device requires two batteries, what is the probability that both batteries last beyond 1 year?

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Problem 20

The length of time of calls made to a support helpline follows an exponential distribution with an average duration of 40 minutes so that $\lambda=1 / 40=.025 .$
a. What is the probability that a call to the helpline lasts less than 5 minutes?
b. What is the probability that a call to the helpline lasts more than 50 minutes?
c. What is the probability that a call lasts between 20 and 30 minutes?
d. Tchebysheff's Theorem says that the interval $40 \pm 2(40)$ should contain at least $75 \%$ of the population. What is the actual probability that the call times lie in this interval?

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