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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

Johnson Electronics makes calculators. Consumer satisfaction is one of the top priorities of the company's management. The company guarantees the refund of money or a replacement for any calculator that malfunctions within two years from the date of purchase. It is known from past data that despite all efforts, \(5 \%\) of the calculators manufactured by this company malfunction within a 2-year period. The company recently mailed 500 such calculators to its customers. a. Find the probability that exactly 29 of the 500 calculators will be returned for refund or replacement within a 2 -year period. b. What is the probability that 27 or more of the 500 calculators will be returned for refund or replacement within a 2 -year period? c. What is the probability that 15 to 22 of the 500 calculators will be returned for refund or replacement within a 2 -year period?

Do the width and/or height of a normal distribution change when its standard deviation remains the same but its mean increases?

Find the area under the standard normal curve a. between \(z=0\) and \(z=1.95\) b. between \(z=0\) and \(z=-2.05\) c. between \(z=1.15\) and \(z=2.37\) d. from \(z=-1.53\) to \(z=-2.88\) e. from \(z=-1.67\) to \(z=2.24\)

The amount of time taken by a bank teller to serve a randomly selected customer has a normal distribution with a mean of 2 minutes and a standard deviation of \(.5\) minute. a. What is the probability that both of two randomly selected customers will take less than 1 minute each to be served? b. What is the probability that at least one of four randomly selected customers will need more than \(2.25\) minutes to be served?

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