Problem 7
Question
Give the range of values that the random variable \(X\) may assume and classify the random variable as finite discrete, infinite discrete, or continuous. \(X=\) The number of times a die is thrown until a 2 appears
Step-by-Step Solution
Verified Answer
The random variable \(X\) represents the number of times a die is thrown until a 2 appears. The minimum value of \(X\) is 1, and there is no maximum value as the die can be thrown indefinitely. The range of values for \(X\) is the set \(\{1, 2, 3, ...\}\). Therefore, \(X\) is an infinite discrete random variable.
1Step 1: Identify the Random Variable
The random variable \(X\) represents the number of times a die is thrown until a 2 appears.
2Step 2: Determine the Minimum Value of \(X\)
The minimum value of \(X\) can be 1, which occurs if a 2 appears on the very first throw.
3Step 3: Determine if the Maximum Value of \(X\) is Finite or Infinite
There is no maximum value of \(X\) as the die can be thrown any number of times until a 2 appears. Theoretically, this could go on indefinitely. Therefore, the maximum value of \(X\) is infinite.
4Step 4: Describe the Range of Values for \(X\)
The range of values for the random variable \(X\) is any positive integer starting from 1, which can be written as the set \(\{1, 2, 3, ...\}\).
5Step 5: Classify the Random Variable
Since the random variable \(X\) is related to rolling a die and its range of values is infinite, we can classify \(X\) as an infinite discrete random variable.
Key Concepts
Discrete Random VariableContinuous Random VariableProbability TheoryMathematical Statistics
Discrete Random Variable
A discrete random variable is a type of random variable that can assume a countable number of distinct values. Think of it like a playlist with a specific number of songs; you can count and list each song. In the given exercise, we are considering the number of times a die is thrown until a 2 appears. This variable is discrete because the outcomes are countable, like ticking off on your fingers 1, 2, 3, and so on for each throw.
In probability theory and mathematical statistics, discrete random variables are usually associated with discrete events. For instance, the flip of a coin (heads or tails), the roll of a die (a number between 1 and 6), and the number of emails you receive in a day are all governed by discrete random variables. What distinguishes them is their inherent limitation to specific values with no in-betweens - you can't roll a 3.5 on a standard die.
In probability theory and mathematical statistics, discrete random variables are usually associated with discrete events. For instance, the flip of a coin (heads or tails), the roll of a die (a number between 1 and 6), and the number of emails you receive in a day are all governed by discrete random variables. What distinguishes them is their inherent limitation to specific values with no in-betweens - you can't roll a 3.5 on a standard die.
Continuous Random Variable
In contrast, a continuous random variable is one that can take on an infinite number of values within a certain range. It's akin to the air in a balloon, which can expand to occupy an infinitely divisible volume. Continuous variables are often measurements, such as height, weight, or time. For instance, the amount of time it takes for a computer to process a task is a continuous random variable—it can be 2 seconds, 2.35 seconds, or any value in between.
When dealing with continuous random variables, we don't count outcomes; we measure them. Unlike discrete random variables, which use probability mass functions (PMFs) to describe their probabilities, continuous random variables require probability density functions (PDFs) due to the uncountable nature of their possible values.
When dealing with continuous random variables, we don't count outcomes; we measure them. Unlike discrete random variables, which use probability mass functions (PMFs) to describe their probabilities, continuous random variables require probability density functions (PDFs) due to the uncountable nature of their possible values.
Probability Theory
At its core, probability theory is the mathematical framework that allows us to analyze random phenomena and make sense of uncertainty. It helps us quantify the likelihood of events, from rolling dice to predicting weather patterns. The exercise we're examining involves probability theory because we're determining the chances of rolling a 2 and how many attempts it might take.
Probability theory works by establishing a set of outcomes and assigning a probability (a number between 0 and 1) to each outcome or set of outcomes. In our die-throwing scenario, probability theory uses concepts such as the probability mass function to characterize the probabilities associated with our discrete random variable, providing us with insights into what we can expect during this random process.
Probability theory works by establishing a set of outcomes and assigning a probability (a number between 0 and 1) to each outcome or set of outcomes. In our die-throwing scenario, probability theory uses concepts such as the probability mass function to characterize the probabilities associated with our discrete random variable, providing us with insights into what we can expect during this random process.
Mathematical Statistics
Moving into the field of mathematical statistics, we take the principles of probability theory and apply them to analyze and interpret data. This branch smartly combines theory and methodology to make inferences about the real world based on samples of data. For example, if you wished to know the average number of times you'd have to roll a die to get a 2, you could use statistical techniques to estimate this number based on repeated trials.
Mathematical statistics allows us to estimate parameters, test hypotheses, and make predictions all based on random variables. It takes the abstract (like our die-throwing game) and makes it applicable, ensuring that we're not just relying on luck when we analyze patterns and trends in the data. In our exercise, techniques from mathematical statistics would enable us to provide a more detailed analysis of the behavior of our random variable across many instances of the experiment.
Mathematical statistics allows us to estimate parameters, test hypotheses, and make predictions all based on random variables. It takes the abstract (like our die-throwing game) and makes it applicable, ensuring that we're not just relying on luck when we analyze patterns and trends in the data. In our exercise, techniques from mathematical statistics would enable us to provide a more detailed analysis of the behavior of our random variable across many instances of the experiment.
Other exercises in this chapter
Problem 6
Determine whether the events \(A\) and \(B\) are independent. \(P(A)=.6, P(B)=.8, P(A \cap B)=.2\)
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Based on past experience, the manager of the VideoRama Store has compiled the following table, which gives the probabilities that a customer who enters the Vide
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If a sample of three batteries is selected from a lot of ten, of which two are defective, what is the expected number of defective batteries?
View solution Problem 8
Give the range of values that the random variable \(X\) may assume and classify the random variable as finite discrete, infinite discrete, or continuous. \(X=\)
View solution