Problem 16 - Applied Statistics and Probability for Engineers

Question

If \(X\) is a continuous random variable, argue that \(P\left(x_{1} \leq X \leq x_{2}\right)=P\left(x_{1}

Step-by-Step Solution

Answer

For continuous random variables, these interval probabilities are equal because the probability of the variable taking on any specific value, \( P(X = x) \), is zero.

Step 1: Understand the Problem

We need to show that the probability expressions for a continuous random variable between two points are equal. This involves showing that the inclusion or exclusion of equality at the boundaries of an interval does not affect the probability for continuous random variables.

Step 2: Properties of Continuous Random Variables

For a continuous random variable, the probability of the variable taking on any specific value is zero. This means that for any value \( x \), \( P(X = x) = 0 \).

Step 3: Set Up the Interval Probabilities

Consider the interval probabilities: \( P(x_{1} \leq X \leq x_{2}) \), \( P(x_{1} < X \leq x_{2}) \), \( P(x_{1} \leq X < x_{2}) \), and \( P(x_{1} < X < x_{2}) \). Although they differ in whether the endpoints are included, we will show this does not impact the probability measure.

Step 4: Analyze Each Probability Expression

For \( P(x_{1} \leq X \leq x_{2}) \), consider it as \( P(x_{1} < X < x_{2}) + P(X = x_1) + P(X = x_2) \). Since \( P(X = x_1) = 0 \) and \( P(X = x_2) = 0 \), we have \( P(x_{1} \leq X \leq x_{2}) = P(x_{1} < X < x_{2}) \).

Step 5: Equate All Interval Probabilities

Similarly, check the other expressions: - \( P(x_{1} < X \leq x_{2}) = P(x_{1} < X < x_{2}) + P(X = x_2)\)- \( P(x_{1} \leq X < x_{2}) = P(x_{1} < X < x_{2}) + P(X = x_1)\)Both \( P(X = x_1) \) and \( P(X = x_2) \) are zero, leading to all expressions being equal to \( P(x_{1} < X < x_{2}) \).

Step 6: Conclude Equivalence

Having analyzed these expressions, we conclude that:\[ P(x_{1} \leq X \leq x_{2}) = P(x_{1} < X \leq x_{2}) = P(x_{1} \leq X < x_{2}) = P(x_{1} < X < x_{2}) \] The interval endpoints' inclusion or exclusion for a continuous random variable does not change the probability.

Key Concepts

Understanding Probability Expressions

When dealing with continuous random variables, probability expressions help us compute the likelihood of the variable occurring within certain boundaries. Unlike discrete random variables, continuous variables can take any value within a range. This means the way boundaries are defined (whether inclusive or exclusive) can create some confusion. The key takeaway is that for continuous variables, all expressions like \( P(x_{1} \leq X \leq x_{2}) \), \( P(x_{1} < X \leq x_{2}) \), \( P(x_{1} \leq X < x_{2}) \), and \( P(x_{1} < X < x_{2}) \) represent the same probability. This happens because the exact value at any specific point has a probability of zero. Thus, the inclusion or exclusion of endpoints in this context does not affect the probability measure. When calculating such probabilities, focus more on the overall range rather than boundary specifics.

Decoding Interval Probabilities

Interval probabilities describe the likelihood that a continuous random variable falls within a specific range of values. They are central in understanding how continuous random variables behave. This concept becomes especially relevant when considering interval notation, as expressed through inequalities. Whether you have closed intervals (\([x_{1}, x_{2}]\)) or open intervals (\((x_{1}, x_{2})\)), which denote the inclusion or exclusion of the boundary points, the interval probability remains constant for continuous variables.

• The interval \([x_1, x_2]\) includes both endpoints \(x_1\) and \(x_2\).
• The interval \((x_1, x_2]\) includes \(x_2\) but excludes \(x_1\).
• The interval \([x_1, x_2)\) includes \(x_1\) but excludes \(x_2\).
• The interval \((x_1, x_2)\) excludes both \(x_1\) and \(x_2\).

Despite these differences, the essence of interval probabilities for continuous variables lies in the understanding that point-specific probabilities, like those at the endpoints, are zero. Therefore, the boundaries' inclusion does not impact the probability that \(X\) falls within the interval.

Essential Properties of Continuous Random Variables

Continuous random variables hold several intrinsic properties that make them unique from discrete random variables. The key characteristic is that they can take any value within a given range, no matter how small the increments. This leads to some fundamental properties:

• Zero Probability at Specific Points: For any given point \( x \), the probability \( P(X = x) = 0 \). This means that the exact probability of observing a particular value is zero.
• Continuous Probability Distribution: Probabilities are calculated over intervals, not individual points. Thus, we compute \( P(a < X < b) \) for ranges \( (a, b) \).
• Probability Density Function (PDF): Continuous variables are often described by a PDF, which outlines the likelihood of random variables falling within a particular range.

Understanding these properties aids in mastering how to work with continuous random variables in probability contexts. They ensure that students approach continuous data with the right mindset, emphasizing ranges rather than discrete values.