CONTINUOUS DISTRIBUTIONS 139 2. Consider the areas shown in Fig. 4.17. In each case, state what probability iS being depicted. What is the relationship between the areas depicted in Figs. 4.17(a) and (b)? Between those in Figs. 4.17(d) and (e)? 3Let X denote the length in minutes of a long-distance telephone conversation. Assume that the density for X is given by f(x) = (1/10)e-*/10 %3D x>0 (a) Verify that f is a density for a continuous random variable. (b) Assuming that f adequately describes the behavior of the random variable X, find the probability that a randomly selected call will last at most 7 minutes; at least 7 minutes; exactly 7 minutes. (c) Would it be unusual for a call to last between 1 and 2 minutes? Explain, based on the probability of this occurring. (d) Sketch the graph of f and indicate in the sketch the area corresponding to each of the probabilities found in part (b). 4.) Some plastics in scrapped cars can be stripped out and broken down to recover the chemical components. The greatest success has been in processing the flex- ible polyurethane cushioning found in these cars. Let X denote the amount of this material, in pounds, found per car. Assume that the density for X is given by 11 f(x) = In 2 x 25 < x< 50 %3D (a) Verify that f is a density for a continuous random variable. (b) Use f to find the probability that a randomly selected auto will contain be- tween 30 and 40 pounds of polyurethane cushioning. (c) Sketch the graph of f, and indicate in the sketch the area corresponding to the probability found in part (b). 5. (Continuous uniform distribution.) A random variable X is said to be uni- formly distributed over an interval (a, b) if its density is given by 1 f(x) : a
CONTINUOUS DISTRIBUTIONS 139 2. Consider the areas shown in Fig. 4.17. In each case, state what probability iS being depicted. What is the relationship between the areas depicted in Figs. 4.17(a) and (b)? Between those in Figs. 4.17(d) and (e)? 3Let X denote the length in minutes of a long-distance telephone conversation. Assume that the density for X is given by f(x) = (1/10)e-*/10 %3D x>0 (a) Verify that f is a density for a continuous random variable. (b) Assuming that f adequately describes the behavior of the random variable X, find the probability that a randomly selected call will last at most 7 minutes; at least 7 minutes; exactly 7 minutes. (c) Would it be unusual for a call to last between 1 and 2 minutes? Explain, based on the probability of this occurring. (d) Sketch the graph of f and indicate in the sketch the area corresponding to each of the probabilities found in part (b). 4.) Some plastics in scrapped cars can be stripped out and broken down to recover the chemical components. The greatest success has been in processing the flex- ible polyurethane cushioning found in these cars. Let X denote the amount of this material, in pounds, found per car. Assume that the density for X is given by 11 f(x) = In 2 x 25 < x< 50 %3D (a) Verify that f is a density for a continuous random variable. (b) Use f to find the probability that a randomly selected auto will contain be- tween 30 and 40 pounds of polyurethane cushioning. (c) Sketch the graph of f, and indicate in the sketch the area corresponding to the probability found in part (b). 5. (Continuous uniform distribution.) A random variable X is said to be uni- formly distributed over an interval (a, b) if its density is given by 1 f(x) : a
A First Course in Probability (10th Edition)
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Compound Probability
Compound probability can be defined as the probability of the two events which are independent. It can be defined as the multiplication of the probability of two events that are not dependent.
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Probability theory is a branch of mathematics that deals with the subject of probability. Although there are many different concepts of probability, probability theory expresses the definition mathematically through a series of axioms. Usually, these axioms express probability in terms of a probability space, which assigns a measure with values ranging from 0 to 1 to a set of outcomes known as the sample space. An event is a subset of these outcomes that is described.
Conditional Probability
By definition, the term probability is expressed as a part of mathematics where the chance of an event that may either occur or not is evaluated and expressed in numerical terms. The range of the value within which probability can be expressed is between 0 and 1. The higher the chance of an event occurring, the closer is its value to be 1. If the probability of an event is 1, it means that the event will happen under all considered circumstances. Similarly, if the probability is exactly 0, then no matter the situation, the event will never occur.
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Transcribed Image Text:CONTINUOUS DISTRIBUTIONS 139
2. Consider the areas shown in Fig. 4.17. In each case, state what probability iS
being depicted. What is the relationship between the areas depicted in Figs.
4.17(a) and (b)? Between those in Figs. 4.17(d) and (e)?
3Let X denote the length in minutes of a long-distance telephone conversation.
Assume that the density for X is given by
f(x) = (1/10)e-*/10
%3D
x>0
(a) Verify that f is a density for a continuous random variable.
(b) Assuming that f adequately describes the behavior of the random variable
X, find the probability that a randomly selected call will last at most
7 minutes; at least 7 minutes; exactly 7 minutes.
(c) Would it be unusual for a call to last between 1 and 2 minutes? Explain,
based on the probability of this occurring.
(d) Sketch the graph of f and indicate in the sketch the area corresponding to
each of the probabilities found in part (b).
4.) Some plastics in scrapped cars can be stripped out and broken down to recover
the chemical components. The greatest success has been in processing the flex-
ible polyurethane cushioning found in these cars. Let X denote the amount of
this material, in pounds, found per car. Assume that the density for X is given by
11
f(x) =
In 2 x
25 < x< 50
%3D
(a) Verify that f is a density for a continuous random variable.
(b) Use f to find the probability that a randomly selected auto will contain be-
tween 30 and 40 pounds of polyurethane cushioning.
(c) Sketch the graph of f, and indicate in the sketch the area corresponding to
the probability found in part (b).
5. (Continuous uniform distribution.) A random variable X is said to be uni-
formly distributed over an interval (a, b) if its density is given by
1
f(x) :
a<x<b
b- a
(a) Show that this is a density for a continuous random variable.
(b) Sketch the graph of the uniform density.
foort ( ) that represents PrY<
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