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Chapter 6: Problem 1

What is the difference between the probability distribution of a discrete random variable and that of a continuous random variable? Explain.

Short Answer

Expert verified
The difference between the probability distributions of discrete and continuous random variables lies in their computation and representation. Discrete random variables use a probability mass function (PMF) which gives direct probabilities for each possible value, whereas continuous random variables use a probability density function (PDF) which provides probabilities over a range of values.

Step by step solution

01

Understanding Discrete Random Variables

A discrete random variable is one that can take on a countable number of distinct values. An example of a discrete random variable could be the number of heads gotten when a coin is tossed three times. This variable can take on values of 0, 1, 2, or 3. The probability distribution of a discrete random variable can be represented by a probability mass function (PMF). The PMF provides the probability of each possible outcome. In the given example, it would give the probabilities for getting 0, 1, 2, or 3 heads.
02

Understanding Continuous Random Variables

A continuous random variable is one that can take on an uncountably infinite number of values. These values could be any number in a given range. An example of a continuous random variable could be the exact height of a randomly chosen individual. The height could be any real number within some range. The probability distribution of a continuous random variable can be represented by a probability density function (PDF). Unlike the PMF, the PDF doesn't give the probability of a single value. Instead, it provides the probability over a range of values. To find the probability that a continuous random variable falls within a certain range, the area under the PDF curve over this range is calculated.
03

The Differences

The primary distinction between a discrete and continuous random variable lies in the way their probabilities are calculated and how their probability distributions are represented. The probability distribution for a discrete random variable is represented by a probability mass function (PMF) and gives direct probabilities for each possible value, while the probability distribution for a continuous random variable is represented by a probability density function (PDF) and gives the probabilities over a range of values.

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Most popular questions from this chapter

The transmission on a model of a specific car has a warranty for 40,000 miles. It is known that the life of such a transmission has a normal distribution with a mean of 72,000 miles and a standard deviation of 13,000 miles. a. What percentage of the transmissions will fail before the end of the warranty period? b. What percentage of the transmissions will be good for more than 100,000 miles?

Under what conditions is the normal distribution usually used as an approximation to the binomial distribution?

According to a May 27, 2009 Minneapolis Star-Tribune article (Source: http://www.startribune.com politics \(/ 45797562 . \mathrm{htm} 1\) ), \(78 \%\) of U.S. households have at least one credit card. Find the probability that in a random sample of 500 U.S. households, 375 to 385 households have at least one credit card

Major League Baseball rules require that the balls used in baseball games must have circumferences between 9 and \(9.25\) inches. Suppose the balls produced by the factory that supplies balls to Major League Baseball have circumferences normally distributed with a mean of \(9.125\) inches and a standard deviation of \(.06\) inch. What percentage of these baseballs fail to meet the circumference requirement?

Determine the \(z\) value for each of the following \(x\) values for a normal distribution with \(\mu=16\) and \(\sigma=3 .\) a. \(x=12\) b. \(x=22\) c. \(x=19\) d. \(x=13\)
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