Problem 11
Question
Let \(X\) be a random variable with distribution function $$ F(x)=\left\\{\begin{array}{ll} 0 & x<-2 \\ 0.2 & -2 \leq x<0 \\ 0.3 & 0 \leq x<1 \\ 0.7 & 1 \leq x<2 \\ 1 & x \geq 2 \end{array}\right. $$ Determine the probability mass function of \(X\).
Step-by-Step Solution
Verified Answer
The probability mass function is: \( P(X = -2) = 0.2 \), \( P(X = 0) = 0.1 \), \( P(X = 1) = 0.4 \), \( P(X = 2) = 0.3 \).
1Step 1: Understand the Cumulative Distribution Function
The given function, \( F(x) \), is a cumulative distribution function (CDF) for the variable \( X \). This function describes the probability that \( X \) will take a value less than or equal to \( x \). The CDF increases only at specific points where the random variable \( X \) can actually take values.
2Step 2: Identify Jump Points
The CDF is a step function and it increases at particular values of \( x \). These increase points are where the probability mass is concentrated. In this case:- \( F(x) \) jumps from 0.2 at \( x = -2 \) to 0.3 at \( x = 0 \),- \( F(x) \) jumps from 0.3 at \( x = 0 \) to 0.7 at \( x = 1 \),- \( F(x) \) jumps from 0.7 at \( x = 1 \) to 1 at \( x = 2 \).
3Step 3: Compute the Probability Mass Function
To find the probability mass function, compute the differences between consecutive steps of the CDF:- At \( x = -2 \), the increase from 0 to 0.2 gives \( P(X = -2) = 0.2 \).- At \( x = 0 \), the increase from 0.2 to 0.3 gives \( P(X = 0) = 0.1 \).- At \( x = 1 \), the increase from 0.3 to 0.7 gives \( P(X = 1) = 0.4 \).- At \( x = 2 \), the increase from 0.7 to 1 gives \( P(X = 2) = 0.3 \).
4Step 4: Verify Total Probability
Ensure that the sum of the probabilities from the pmf is equal to 1:\[ P(X = -2) + P(X = 0) + P(X = 1) + P(X = 2) = 0.2 + 0.1 + 0.4 + 0.3 = 1.0 \]This confirms that the probability mass function is correctly identified.
Key Concepts
Cumulative Distribution FunctionRandom VariableProbability Theory
Cumulative Distribution Function
A Cumulative Distribution Function (CDF) is a fundamental concept in probability theory. It is used to describe the distribution of a random variable. The CDF, denoted by \( F(x) \), provides the probability that a random variable \( X \) is less than or equal to a certain value \( x \). The CDF is a step function in discrete variables and a continuous curve in continuous variables.
For example, let's consider the random variable \( X \) with the provided distribution function:
\[ \ F(x)=\left\{\begin{array}{ll} 0 & x<-2 \ 0.2 & -2 \leq x<0 \ 0.3 & 0 \leq x<1 \ 0.7 & 1 \leq x<2 \ 1 & x \geq 2 \end{array}\right. \]
In this context, \( F(x) \) describes the probability distribution in incremental steps. It only increases at specific values, indicating where the random variable \( X \) can have non-zero probabilities, which is essential for creating another function known as the probability mass function (PMF). Understanding the CDF is crucial because it lays the groundwork for deriving the PMF by indicating which values contribute to the total probability.
For example, let's consider the random variable \( X \) with the provided distribution function:
\[ \ F(x)=\left\{\begin{array}{ll} 0 & x<-2 \ 0.2 & -2 \leq x<0 \ 0.3 & 0 \leq x<1 \ 0.7 & 1 \leq x<2 \ 1 & x \geq 2 \end{array}\right. \]
In this context, \( F(x) \) describes the probability distribution in incremental steps. It only increases at specific values, indicating where the random variable \( X \) can have non-zero probabilities, which is essential for creating another function known as the probability mass function (PMF). Understanding the CDF is crucial because it lays the groundwork for deriving the PMF by indicating which values contribute to the total probability.
Random Variable
A random variable is a variable whose values are outcomes of a random phenomenon.
The concept is to quantify outcomes from random processes, allowing us to assign numerical values to different events. There are two main types of random variables: discrete and continuous. Each serves a different purpose depending on whether the dataset or scenario has gaps between potential values or not.
In our example, \( X \) is a discrete random variable. It takes specific values such as \( -2, 0, 1, \) and \( 2 \). The behavior of \( X \) can be studied using a Cumulative Distribution Function to see the likelihood of \( X \) being less than or equal to any value within its range. Notably, this function shows the points where \( X \) has defined probabilities leading to a clearer understanding of the underlying probability structure.
Random variables simplify statistical calculations and modeling complex probability systems, where outcomes aren't just random but quantifiable.
The concept is to quantify outcomes from random processes, allowing us to assign numerical values to different events. There are two main types of random variables: discrete and continuous. Each serves a different purpose depending on whether the dataset or scenario has gaps between potential values or not.
In our example, \( X \) is a discrete random variable. It takes specific values such as \( -2, 0, 1, \) and \( 2 \). The behavior of \( X \) can be studied using a Cumulative Distribution Function to see the likelihood of \( X \) being less than or equal to any value within its range. Notably, this function shows the points where \( X \) has defined probabilities leading to a clearer understanding of the underlying probability structure.
Random variables simplify statistical calculations and modeling complex probability systems, where outcomes aren't just random but quantifiable.
Probability Theory
Probability Theory is the mathematical framework for quantifying uncertainty. It provides the tools and principles necessary to calculate the likelihood of events occurring.
One of the key ideas in probability theory is the understanding of events through distributions like the Cumulative Distribution Function (CDF) and functions like the Probability Mass Function (PMF). These functions help break down complex stochastic systems into digestible parts and are critical tools in assessing scenarios involving randomness.
The process of transitioning from a CDF to a PMF, as outlined in the exercise, is a practical demonstration of probability theory principles. By examining where and how frequently certain outcomes occur, we can compile probabilities into a PMF that accurately represents the possible outcomes and ensures the total probability sums to one.
Probability Theory helps in a wide array of applications from gambling strategies, decision-making processes, risk assessment, to machine learning algorithms.
One of the key ideas in probability theory is the understanding of events through distributions like the Cumulative Distribution Function (CDF) and functions like the Probability Mass Function (PMF). These functions help break down complex stochastic systems into digestible parts and are critical tools in assessing scenarios involving randomness.
The process of transitioning from a CDF to a PMF, as outlined in the exercise, is a practical demonstration of probability theory principles. By examining where and how frequently certain outcomes occur, we can compile probabilities into a PMF that accurately represents the possible outcomes and ensures the total probability sums to one.
Probability Theory helps in a wide array of applications from gambling strategies, decision-making processes, risk assessment, to machine learning algorithms.
Other exercises in this chapter
Problem 11
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