Question

Let the probability set function of the random variable \(X\) be $$P_{X}(C)=\int_{C} e^{-x} d x, \quad \text { where } \mathcal{C}=\\{x: 0

Short answer

\(\lim_{k \rightarrow \infty} C_k = [2,3]\), \(P_X(\lim_{k \rightarrow \infty} C_k) = -e^{-3} + e^{-2}\), and \(\lim_{k \rightarrow \infty} P_X(C_k) = -e^{-3} + e^{-2}\). Therefore, the statement \(\lim_{k \rightarrow \infty} P_X(C_k) = P_X(\lim_{k \rightarrow \infty} C_k)\) holds true.

Step-by-step solution

Find the limit of the sequence \(C_k\)

The sequence \(C_k\) converges as \(k \rightarrow \infty\), because \((2 -1/k)\) approaches 2 and is always less than or equal to 3, meaning \(C_k = \{x: 2 < x \leq 3\}\). Thus, \(\lim_{k \rightarrow \infty} C_k = [2,3]\).

Evaluate the probability function \(P_X(C)\)

The probability function evaluates over the integral of the provided set. For our set function \(C_k\), we need to calculate \(\int_{2}^{3} e^{-x} dx\), which will give us the value of \(P_X(C)\). By using the properties of the exponential function, this evaluates to \(-e^{-3} + e^{-2}\).

Evaluate \(P_X(\lim_{k \rightarrow \infty} C_k)\)

We already found that \(\lim_{k \rightarrow \infty} C_k = [2,3]\). So we simply substitute [2,3] for C in the probability function to get: \(P_X([2,3]) = -e^{-3} + e^{-2}\).

Find \(P_X(C_k)\) and its limit as \(k \rightarrow \infty\)

To find \(P_X(C_k)\), we need to evaluate the integral \(\int_{2 - 1/k}^{3} e^{-x} dx\). This is the probability function over the interval \(C_k\). Simplifying this will give us \(-e^{-3} + e^{-(2 - 1/k)}\). As \(k \rightarrow \infty\), \(1/k \rightarrow 0\), hence the limit of \(P_X(C_k)\) as \(k \rightarrow \infty\) equals to \(P_X(\lim_{k \rightarrow \infty} C_k) = -e^{-3} + e^{-2}\).

Compare the results

We know from steps 3 and 4 that \(P_X(\lim_{k \rightarrow \infty} C_k) = -e^{-3} + e^{-2}\) and \(\lim_{k \rightarrow \infty} P_X(C_k) = -e^{-3} + e^{-2}\). They are equal, confirming our solution is correct.

Key concepts

Understanding Random Variables

Random variables are fundamental in probability theory as they help explain and model phenomena with uncertain outcomes. A random variable, typically denoted as \( X \), is a variable that takes on possible outcomes in a random manner, often mapped to real numbers. These outcomes are determined by some probability distribution.
For instance, if you flip a coin, the outcome of heads or tails can be represented by a random variable \( X \) where heads could be assigned a value of 1 and tails a value of 0. Thus, random variables provide a way to quantify uncertainty and make calculations in probability quite flexible and insightful.

• Discrete Random Variables: These have countable outcomes, like the result of a dice throw.
• Continuous Random Variables: These can take any value within a range, such as the exact height of students in a class.

In the given exercise, the random variable \( X \) has an associated probability distribution given by an integral, which leads us to our next topic.

Exploring Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between events in a Poisson process. It is defined by its probability density function (PDF), which for a given rate \( \lambda \) is \( f(x) = \lambda e^{-\lambda x} \) for \( x \geq 0 \).
This distribution is memoryless, meaning the probability of an event occurring in the future is independent of past events. This unique property makes the exponential distribution particularly useful in fields like queuing theory, reliability engineering, and survival analysis.

• The mean of an exponential distribution is given by \( 1/\lambda \).
• The variance is also given by \( 1/\lambda^2 \).

For the problem at hand, the random variable \( X \) follows an exponential distribution with a rate parameter \( \lambda = 1 \), as indicated by the probability set function \( P_X(C) = \int_C e^{-x} \, dx \). Calculating integrals within certain limits helps find the probability for variable \( X \) over defined ranges, such as \( C_k \).

Understanding the Convergence of Sets

The convergence of sets is an important concept when dealing with sequences of sets in probability theory and analysis. It involves studying how a sequence of sets behaves as an index approaches infinity, essentially when the set members settle into a limit set. A sequence of sets \( \{C_k\} \) converges if it approaches a limiting set \( C \) as \( k \to \infty \).
In this exercise, \( C_k = \{x : 2 - 1/k < x \leq 3\} \) describes an interval on the real line that evolves as \( k \) increases.

• As \( k \rightarrow \infty \), the term \( 1/k \) shrinks, making the left boundary of \( C_k \) approach 2, thus forming the limiting interval \([2, 3]\).
• The probability measure of the resulting set \( [2, 3] \) can be calculated using the provided integral of the exponential function.

Understanding set convergence helps predict and analyze the behavior of probabilities over changing conditions, a crucial skill in advanced calculus and probability theory. The solution confirms that \( \lim_{k \rightarrow \infty} P_X(C_k) = P_X(\lim_{k \rightarrow \infty} C_k) \), showcasing the power of integrating set convergence with probability.