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Chapter 6: Problem 3

For a continuous probability distribution, explain why the following holds true. $$ P(a

Short Answer

Expert verified
For a continuous probability distribution, the probabilities P(a

Step by step solution

01

Understanding a Continuous Probability Distribution

In a continuous probability distribution, the concerned random variable can take on any value within a range. This can be any interval of real numbers, either finite or infinite. The probabilities of individual points are all zero in continuous probability distributions because there are infinite possibilities.
02

The implication of this property for event probabilities

If the probabilities of individual points are all zero, then when considering events that are ranges of values, adding or not a single point makes no substantial difference. This means that the probability of a random variable falling within a certain range is not affected by whether or not we include the endpoints of the range. Specifically, the probability that a random variable X takes on values between a and b is the same, regardless of whether or not we include a and/or b.
03

Equating given probabilities

By applying these concepts, we can confirm that P(a

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Most popular questions from this chapter

The average monthly mortgage payment for all homeowners in a city is $$\$ 2850.$$ Suppose that the distribution of monthly mortgages paid by homeowners in this city follow an approximate normal distribution with a mean of $$\$ 2850$$ and a standard deviation of $$\$ 420.$$ Find the probability that the monthly mortgage paid by a randomly selected homeowner from this city is a. less than $$\$ 1200$$ b. between $$\$ 2300$$ and $$\$ 3140$$ c. more than $$\$ 3600$$ d. between $$\$ 3200$$ and $$\$ 3700$$

How do the width and height of a normal distribution change when its mean remains the same but its standard deviation decreases?

For the standard normal distribution, find the area within one standard deviation of the mean - that is, the area between \(\mu-\sigma\) and \(\mu+\sigma .\)

For a binomial probability distribution, \(n=25\) and \(p=.40\). a. Find the probability \(P(8 \leq x \leq 13)\) by using the table of binomial probabilities (Table I of Appendix B). b. Find the probability \(P(8 \leq x \leq 13)\) by using the normal distribution as an approximation to the binomial distribution. What is the difference between this approximation and the exact probability calculated in part a?

A construction zone on a highway has a posted speed limit of 40 miles per hour. The speeds of vehicles passing through this construction zone are normally distributed with a mean of 46 miles per hour and a standard deviation of 4 miles per hour. Find the percentage of vehicles passing through this construction zone that are a. exceeding the posted speed limit b. traveling at speeds between 50 and 57 miles per hour
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