Exponential distribution question

In summary, in the first page, the steps are arrived at by applying the limit of a function and defining $R_X(y)$ as an intermediate step. $R_X(y)$ is related to $F_X(y)$ and its derivative, $R'_X(y)$. The solution to the differential equation is then obtained using $R_X(y)$ and its derivative, and it is different from the original equation. The derivative of the cumulative distribution function is the probability density function, but in this case it is difficult to understand.
  • #1
WMDhamnekar
MHB
376
28
Hi,
1594715485258.png
1594715729229.png


I want to know how the highlighted steps are arrived at in the first page. What are \(R_X (y), R'_X (y),F'_X (0) ? \)How \(R_X (0) = 1 ?\) Solution to differential equation should be \(R_X (y)=K*e^{\int{R'_X (0) dx}}\) But it is different. How is that?

What is $-R'_X (0)=F'_X(0)=f_X(0)$ I know derivative of CDF is PDF, but in this case it is somewhat difficult to understand.

If any member of MHB knows how to satisfy my queries correctly, may reply to this question:confused::unsure:
 
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  • #2
Hey Dhamnekar Winod,

In the first step they apply:
$$\lim_{t\to 0} \frac{F_X(t)[1-F_X(y)]}{t} = \lim_{t\to 0} \frac{F_X(t)-F_X(0)}{t}\cdot [1-F_X(y)] = F_X'(0)\cdot [1-F_X(y)]$$

Apparently they defined $R_X(y)=1-F_X(y)$ as an intermediate step to solve the differential equation.
It follows that $R_X'(y)=-F_X'(y)$ and $R_X(0)=1$.
The corresponding equation then follows from the previous equation.

The solution of the differential equation is indeed:
$$R_X (y)=K\cdot e^{\displaystyle\int_0^y{R'_X (0) dx}} = K\cdot e^{\displaystyle\big[R'_X (0) x\big]_0^y} = K\cdot e^{R'_X (0) y}$$
 

1. What is the Exponential distribution?

The Exponential distribution is a probability distribution that models the time between events occurring at a constant rate. It is often used to model the time between arrivals in a queuing system or the lifetime of a product.

2. How is the Exponential distribution different from other probability distributions?

The Exponential distribution is unique in that it is the only continuous probability distribution that has a constant hazard rate (the probability of an event occurring at any given time is always the same). This makes it useful for modeling processes that are memoryless, meaning the probability of an event occurring is independent of how much time has passed.

3. What are the key properties of the Exponential distribution?

The key properties of the Exponential distribution include its shape, which is a right-skewed curve with a long tail, and its mean and standard deviation, which are both equal to the inverse of the rate parameter (λ). It also has a cumulative distribution function (CDF) that is used to calculate the probability of an event occurring within a certain time frame.

4. How is the Exponential distribution used in real-world applications?

The Exponential distribution has a wide range of applications in various fields, including engineering, finance, and healthcare. It is commonly used to model the time between equipment failures, the time between customer arrivals in a service system, and the time between occurrences of rare events such as natural disasters. It is also used in survival analysis to model the time until an event, such as death, occurs.

5. What are some common misconceptions about the Exponential distribution?

One common misconception is that the Exponential distribution can be used to model any process with a constant rate. In reality, it is only appropriate for processes that are memoryless. Another misconception is that the Exponential distribution can be used to model discrete events, when in fact it is a continuous distribution. It is important to carefully consider the assumptions and limitations of the Exponential distribution before applying it to a real-world problem.

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