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Chapter 8: Problem 62

If \(X\) is a continuous random variable, what values can the quantity \(P(X=a)\) have?

Short Answer

Expert verified
For a continuous random variable \(X\), the probability of the variable taking a specific value (\(P(X=a)\)) is always 0, as the probability distribution is continuous and not concentrated on individual points.

Step by step solution

01

Understand Continuous Random Variables

A continuous random variable can take any value within a given range or interval (unlike discrete random variables, which can only take certain specific values). Because of this, the probability distribution function (PDF) of a continuous random variable is a continuous function over the interval.
02

Probability Distribution Function (PDF)

To find the probability of a continuous random variable taking a value within a specified interval, we use the area under the curve of the probability distribution function (PDF). The total area under the curve of the PDF should equal to 1, as it represents the sum of all possible probabilities for the variable.
03

Probability of a Specific Value

For a continuous random variable, it's important to note that no specific value of the variable has a nonzero probability. Since there's an infinite number of possible values, the probability of a specific value (in this case, \(P(X=a)\)) is always 0. In conclusion, for a continuous random variable (X), the probability of the variable taking a specific value (\(P(X=a)\)) is always 0, as the probability distribution is continuous and not concentrated on individual points.

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Most popular questions from this chapter

A survey of 52 U.S. supermarkets yielded the following relative frequency table, where \(X\) is the number of checkout lanes at a randomly chosen supermarket. \({ }^{23}\) $$ \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) & .01 & .04 & .04 & .08 & .10 & .15 & .25 & .20 & .08 & .05 \\ \hline \end{array} $$ a. Compute \(\mu=E(X)\) and interpret the result. HINT [See Example 3.] b. Which is larger, \(P(X<\mu)\) or \(P(X>\mu)\) ? Interpret the result.

Video Arcades Your company, Sonic Video, Inc., has conducted research that shows the following probability distribution, where \(X\) is the number of video arcades in a randomly chosen city with more than 500,000 inhabitants: \begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|} \hline\(x\) & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline\(P(X=x)\) & \(.07\) & \(.09\) & \(.35\) & \(.25\) & \(.15\) & \(.03\) & \(.02\) & \(.02\) & \(.01\) & \(.01\) \\ \hline \end{tabular} a. Compute the mean, variance, and standard deviation (accurate to one decimal place). b. As CEO of Startrooper Video Unlimited, you wish to install a chain of video arcades in Sleepy City, U.S.A. The city council regulations require that the number of arcades be within the range shared by at least \(75 \%\) of all cities. What is this range? What is the largest number of video arcades you should install so as to comply with this regulation?

Calculate the expected value, the variance, and the standard deviation of the given random variable \(X .\) You calculated the expected values in the last exercise set. Round all answers to two decimal places.) Thirty darts are thrown at a dartboard. The probability of hitting a bull's-eye is \(\frac{1}{5}\). Let \(X\) be the number of bull's-eyes hit.

The following table shows tow ratings (in pounds) for some popular sports utility vehicles: \({ }^{5}\) \begin{tabular}{|l|l|} \hline Mercedes Grand Marquis V8 & 2,000 \\ \hline Jeep Wrangler I6 & 2,000 \\ \hline Ford Explorer V6 & 3,000 \\ \hline Dodge Dakota V6 & 4,000 \\ \hline Mitsubishi Montero V6 & 5,000 \\ \hline Ford Explorer V8 & 6,000 \\ \hline Dodge Durango V8 & 6,000 \\ \hline Dodge Ram 1500 V8 & 8,000 \\ \hline Ford Expedition V8 & 8,000 \\ \hline Hummer 2-door Hardtop & 8,000 \\ \hline \end{tabular} Let \(X\) be the tow rating of a randomly chosen popular SUV from the list above. a. What are the values of \(X ?\) b. Compute the frequency and probability distributions of c. What is the probability that an SUV (from the list above) is rated to tow no more than 5,000 pounds?

If we define a "rich" household as one whose after-tax income is at least \(1.3\) standard deviations above the mean, what is the household income of a rich family in the United States?
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