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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<x<b), P(a<x<=b), P(a<=x<b) and P(a<=x<=b) are all equal because in such distributions, the probability of any single point is 0. Therefore, including or excluding an endpoint of an interval doesn't affect the calculated probability.

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<x<b)=P(a<x<=b)=P(a<=x<b)=P(a<=x<=b), as including or not including the endpoints ('a' or 'b') does not affect the final calculated probability in a continuous probability distribution. This follows directly from the properties of continuous probabilities.

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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 $$\$ 3700$$

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

A variation of a roulette wheel has slots that are not of equal size. Instead, the width of any slot is proportional to the probability that a standard normal random variable \(z\) takes on a value between \(a\) and \((a+.1)\), where \(a=-3.0,-2.9,-2.8, \ldots, 2.9,3.0 .\) In other words, there are slots for the intervals \((-3.0,-2.9),(-2.9,-2.8)\), \((-2.8,-2.7)\) through \((2.9,3.0)\). There is one more slot that represents the probability that \(z\) falls outside the interval \((-3.0,3.0)\). Find the following probabilities. a. The ball lands in the slot representing \((.3, .4)\). b. The ball lands in any of the slots representing \((-.1, .4)\). c. In at least one out of five games, the ball lands in the slot representing \((-.1, .4)\). d. In at least 100 out of 500 games, the ball lands in the slot representing \((.4, .5)\)

For the standard normal distribution, what does \(z\) represent?

One of the cars sold by Walt's car dealership is a very popular subcompact car called the Rhino. The final sale price of the basic model of this car varies from customer to customer depending on the negotiating skills and persistence of the customer. Assume that these sale prices of this car are normally distributed with a mean of $$\$ 19,800$$ and a standard deviation of $$\$ 350.$$ a. Dolores paid $$\$ 19,445$$ for her Rhino. What percentage of Walt's customers paid less than Dolores for a Rhino? b. Cuthbert paid $$\$ 20,300$$ for a Rhino. What percentage of Walt's customers paid more than Cuthbert for a Rhino?
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