Question

If \(X_{1}\) and \(X_{2}\) are independent nonnegative continuous random variables, show that $$ P\left\\{X_{1}

Short answer

In order to prove \(P\left\{X_{1}<X_{2} \mid \min \left(X_{1}, X_{2}\right)=t\right\}=\frac{r_{1}(t)}{r_{1}(t)+r_{2}(t)}\), we expressed the probability of the given condition in terms of the distributions of the random variables \(X_1\) and \(X_2\), and utilized the properties of the failure rate function. After breaking down the numerator and denominator and substituting the failure rate function expressions, we simplified the conditional probability and found that the given relationship holds true.

Step-by-step solution

Define the given probability

We are asked to prove: \( P\left\{X_{1}<X_{2} \mid \min \left(X_{1}, X_{2}\right)=t\right\}=\frac{r_{1}(t)}{r_{1}(t)+r_{2}(t)} \) Where \(X_1\) and \(X_2\) are independent nonnegative continuous random variables, and \(r_i(t)\) is the failure rate function for \(X_i\).

Express the probability of the given condition

We know that \(P\left\{X_{1}<X_{2} \mid \min \left(X_{1}, X_{2}\right)=t\right\} = \frac{P\left\{X_1 < X_2, \min(X_1, X_2) = t \right\}}{P\left\{\min(X_1, X_2) = t\right\}}\)

Break down the numerator

We have \(P\left\{X_1 < X_2, \min(X_1, X_2) = t\right\}= P\left\{X_1 = t, X_2 > t\right\}\) since \(\min(X_1, X_2) = t\) and \(X_1 < X_2\). Since \(X_1\) and \(X_2\) are independent, we can write the joint probability as a product of their densities: \(P\left\{X_1 = t, X_2 > t\right\} = f_1(t) \int_t^{\infty} f_2(x_2) dx_2\) Where \(f_1(t)\) and \(f_2(x_2)\) are the probability density functions of \(X_1\) and \(X_2\), respectively.

Break down the denominator

The denominator can be expressed as: \(P\left\{\min(X_1, X_2) = t\right\} = P\left\{X_1 = t, X_2 > t\right\} + P\left\{X_2 = t, X_1 > t\right\}\) Like the numerator, we can break down the probabilities into a product of their densities: \(P\left\{\min(X_1, X_2) = t\right\} = f_1(t) \int_t^{\infty} f_2(x_2) dx_2 + f_2(t) \int_t^{\infty} f_1(x_1) dx_1\)

Compute the failure rate function

The failure rate function \(r_i(t)\) for \(X_i\) is given by: \(r_i(t) = \frac{f_i(t)}{1 - F_i(t)}\) Where \(F_i(t)\) is the cumulative distribution function of \(X_i\). To relate this with our expressions in steps 3 and 4, we need to manipulate this expression to include our densities. We can rewrite the expression as: \(f_i(t) = r_i(t)(1 - F_i(t))\)

Substitute the failure rate function into our expressions

Using the rewritten failure rate functions in Steps 3 and 4 expressions: Numerator: \(P\left\{X_1 = t, X_2 > t\right\} = r_1(t)(1 - F_1(t)) \int_t^{\infty} f_2(x_2) dx_2\) Denominator: \(P\left\{\min(X_1, X_2) = t\right\} = r_1(t)(1 - F_1(t)) \int_t^{\infty} f_2(x_2) dx_2 + r_2(t)(1 - F_2(t)) \int_t^{\infty} f_1(x_1) dx_1\)

Substitute and simplify the conditional probability

Substitute the expressions we obtained so far back into the original conditional probability expression: \( P\left\{X_{1}<X_{2} \mid \min \left(X_{1}, X_{2}\right)=t\right\}=\frac{r_1(t)(1 - F_1(t)) \int_t^{\infty} f_2(x_2) dx_2}{r_1(t)(1 - F_1(t)) \int_t^{\infty} f_2(x_2) dx_2 + r_2(t)(1 - F_2(t)) \int_t^{\infty} f_1(x_1) dx_1} \) Now, notice that the terms \((1 - F_1(t)) \int_t^{\infty} f_2(x_2) dx_2\) in the numerator and denominator cancel out, leaving us with: \( P\left\{X_{1}<X_{2} \mid \min \left(X_{1}, X_{2}\right)=t\right\}=\frac{r_1(t)}{r_1(t)+r_2(t)} \) This completes the proof.

Key concepts

Continuous Random Variables

When exploring the realm of probability, we encounter continuous random variables. Unlike their discrete counterparts, which take on distinct values, continuous random variables can take on any value within a certain range or interval. This range may be finite or infinite, and the variables are often associated with measurements like time, weight, or distance.

To illustrate, think of measuring the length of a leaf. It could be any number from, say, 5 to 10 centimeters, including any fraction in between. This is a continuous range, and if the leaf length is a random variable, it would be a continuous random variable.

Failure Rate Function

The concept of a failure rate function enters the stage particularly in the context of survival analysis or reliability engineering. It describes the rate at which failures occur over time. Formally, for a continuous random variable representing the time until failure, the failure rate function, often denoted by \(r(t)\), is defined as the ratio of the probability density function (PDF) of the time to failure at a specific time \(t\) to the probability of surviving until that time, expressed by the complementary cumulative distribution function (CCDF).

Mathematical Expression

The mathematical expression for the failure rate function is \(r(t) = \frac{f(t)}{1 - F(t)}\), where \(f(t)\) is the PDF and \(F(t)\) is the cumulative distribution function (CDF). This ratio gives insight into how likely a failure is to occur at a particular moment, given that it hasn't occurred yet.

Probability Density Function

Drill down into probability density function (PDF), and we find the cornerstone for working with continuous random variables. The PDF, denoted as \(f(x)\), represents the likelihood of the random variable falling within a particular infinitesimal range near \(x\).

Contrary to probability mass functions for discrete variables which give probabilities directly, the PDF itself is not a probability. Instead, probabilities are determined through integration over an interval. If you want to know the probability that a variable is between \(a\) and \(b\), you integrate the PDF over that range. Therefore, the area under the entire PDF curve over its range (which is always positive) sums up to 1, signifying the total probability.

Cumulative Distribution Function

Threading further down the statistical path, we encounter the cumulative distribution function (CDF). It is related to the PDF but with a distinct purpose — the CDF, denoted by \(F(x)\), tells us the probability that a continuous random variable will take a value less than or equal to \(x\).

Imagine rolling up the area under the PDF curve from negative infinity to a point \(x\); the CDF reflects this accumulated probability. It's a non-decreasing function that starts off at 0 and approaches 1 as \(x\) heads towards infinity. The CDF is an essential tool for understanding the overall distribution and behavior of random variables.

Independent Random Variables

The concept of independent random variables is vital when dealing with multiple stochastic processes. Independence implies that the occurrence of one random event does not influence the probability of occurrence of another. To put it another way, knowing the outcome of one doesn't provide any information about the other.

In our exercise, we are dealing with two independent random variables \(X_1\) and \(X_2\). This allowed us to express the probability of a joint event as the product of the individual probabilities or PDFs for each variable. Independence is a powerful assumption that simplifies complex probability calculations, keeping each variable's behavior strictly within its own domain, unaffected by its counterparts.