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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 distribution of a discrete and a continuous random variable is that a discrete random variable is represented by a probability mass function (PMF), which assigns probabilities to exact values, while a continuous random variable is represented by a probability density function (PDF), where probabilities are assigned over an interval of values.

Step by step solution

01

- Understand Discrete Random Variable

A discrete random variable is one which may take on only a countable number of distinct values such as 0,1,2,3,4,... etc. It can be visualized on a graph where the X-axis represents the outcomes and the Y-axis represents the probability of the outcomes. A discrete probability distribution can be represented in a probability mass function (PMF). The PMF is a function that gives the probability that a discrete random variable is exactly equal to some value.
02

- Understand Continuous Random Variable

A continuous random variable is one which takes an infinite number of possible values. It's a random variable with a set of possible values, known as the range, that is infinite and uncountable. A continuous random variable is not defined at specific values. Instead, it is defined over an interval of values, and is represented by the area under a curve (the integral). A continuous probability distribution can be described by a probability density function (PDF).
03

- Differentiate Between Discrete and Continuous Random Variables

The main difference between these two kinds of random variables lies in the set of possible outcomes. While a discrete random variable can have a countable number of outcomes, a continuous random variable has an uncountable number of outcomes. Furthermore, the probability distribution of a discrete random variable is represented by a PMF, which assigns probabilities to exact values, while the probability distribution of a continuous random variable is represented by a PDF, where probabilities are assigned over an interval of values.

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Most popular questions from this chapter

A machine at Kasem Steel Corporation makes iron rods that are supposed to be 50 inches long. However, the machine does not make all rods of exactly the same length. It is known that the probability distribution of the lengths of rods made on this machine is normal with a mean of 50 inches and a standard deviation of \(.06\) inch. The rods that are either shorter than \(49.85\) inches or longer than \(50.15\) inches are discarded. What percentage of the rods made on this machine are discarded?

Let \(x\) be a continuous random variable. What is the probability that \(x\) assumes a single value, such as \(a\) ?

According to a survey, \(15 \%\) of U.S. adults with online services currently read e-books. Assume that this percentage is true for the current population of U.S. adults with online services. Find the probability that in a random sample of 600 U.S. adults with online services, the number who read e-books is a. exactly 97 b. at most 106 c. 76 to 99

Suppose you are conducting a binomial experiment that has 15 trials and the probability of success of \(.02\). According to the sample size requirements, you cannot use the normal distribution to approximate the binomial distribution in this situation. Use the mean and standard deviation of this binomial distribution and the empirical rule to explain why there is a problem in this situation. (Note: Drawing the graph and marking the values that correspond to the empirical rule is a good way to start.)

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 \((.4, .5)\)
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