Let X and Y be independent random variables with X ∼θ and Y ∼ρ for certain θ, ρ>0. Show that 𝐏[X

Let X and Y be independent random variables with X ∼θ and Y...

Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between independent events that happen at a constant average rate. Its probability density function is defined by a single parameter (often called the rate), which governs how rapidly events occur. It is characterized by its memoryless property, meaning that the probability of an event occurring in the next period is independent of how long one has already waited.

Independence of Random Variables

Independence between random variables implies that the occurrence or value of one does not affect the probability distribution of the other. This characteristic allows the joint probability density function of two independent variables to be expressed as the product of their individual densities. This property is crucial for calculating probabilities involving both variables, such as evaluating the likelihood that one variable is less than another.

Competing Risks and Order Statistics

The scenario where two independently occurring events (each modeled by an exponential distribution) 'compete' to happen first is often referred to as a competing risks problem. In such cases, the probability that one event occurs before the other is determined by the respective rates of the exponential distributions. By integrating over the possible outcomes using the memoryless and independence properties, one can show that the probability that one variable is smaller than the other is given by the ratio of its rate to the sum of both rates.

Transcript

00:01 In this question, we know that since x, y are exponential distributions, so we have the pdf should be theta, exponential negative theta x, and the pdf of y should be rho, exponential negative rho y.

00:33 We know that x, y are independent, so the joint pdf, this should be the product.

00:59 So it is theta rho, exponential negative theta x, exponential negative rho y.

01:11 Now we are gonna compute the probability that x is less than y.

01:23 This is what we are gonna compute.

01:25 So we can draw the graph of this region.

01:33 Since we know that we're only considering the x, y greater than or equal to zero, because this is exponential distribution, it's supported on the positive line.

01:51 So we only consider the first quadrant.

01:55 This line is x equal y.

02:01 And x less than y is this region.

02:05 So we are going to compute the integral.

02:15 We let y from zero to infinity.

02:21 So dy zero to infinity.

02:24 And for each y, x goes from zero to y.

02:29 And the integral is this pdf.

02:33 So this is integral from zero to infinity, integral from zero to y, theta rho, exponential negative theta x, exponential negative rho y dx dy.

02:52 We first compute the inner integral with respect to x.

02:58 So theta rho, exponential negative rho y.

03:02 They don't have x, so we view it as constant when integrating with respect to x.