How to find PDF and Expected value of max(x,0), for a random variable x

In summary, the PDF for $E$ can be found by transforming the PDF of the exponential distribution and the expected value of $E$ can be found by integrating the PDF of $E$ over its support.
  • #1
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Let $a,b,c, \tau$ be positive constants and $x$ is an exponentially distributed variable with parameter $\lambda = 1$, i.e. $x\sim\exp(1)$.

\begin{equation}
E = \tau\Big[a\frac{1+a}{1+e^{-bx+c}} - 1 \Big]^+
\end{equation}


where $[z]^+ = \max(z,0)$

How can I find

  1. The PDF for $E$
  2. The expected value of E.
 
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  • #2
The PDF for $E$ can be found by using the probability density function of the exponential distribution and transforming it into a function for $E$. The PDF of $E$ can then be expressed as:$$f_E(x) = \tau\Big[a\frac{1+a}{1+e^{-bx+c}} - 1 \Big]^+ \cdot \lambda e^{-\lambda x}$$The expected value of $E$ can then be found by integrating the PDF of $E$ over its support. $$E[E] = \int_{-\infty}^{\infty} \tau\Big[a\frac{1+a}{1+e^{-bx+c}} - 1 \Big]^+ \cdot \lambda e^{-\lambda x}\,dx $$This integral can be solved using integration by parts.
 

Related to How to find PDF and Expected value of max(x,0), for a random variable x

1. What is a PDF and how is it related to a random variable?

A PDF, or probability density function, is a mathematical function that describes the probability of a random variable taking on a certain value. In other words, it shows the likelihood of a specific outcome occurring for a given random variable.

2. How do you find the PDF of a random variable?

The PDF of a random variable x can be found by taking the derivative of its cumulative distribution function (CDF). The CDF is a function that gives the probability that a random variable is less than or equal to a specific value. By taking the derivative, we can find the rate of change of the CDF, which gives us the PDF.

3. What is the expected value of a random variable?

The expected value, also known as the mean or average, of a random variable is the sum of all possible outcomes multiplied by their respective probabilities. It represents the long-term average value of the random variable.

4. How do you find the expected value of max(x,0) for a random variable x?

To find the expected value of max(x,0) for a random variable x, we first need to find the CDF of max(x,0). Then, we can take the derivative of the CDF to find the PDF. Finally, we can use the formula for expected value to calculate the average value of max(x,0).

5. Why is it important to find the PDF and expected value of max(x,0) for a random variable x?

Knowing the PDF and expected value of max(x,0) for a random variable x allows us to understand the behavior and characteristics of the random variable. It can help us make predictions and decisions based on the likelihood of certain outcomes occurring. Additionally, it is a fundamental concept in probability and statistics, which are essential in many scientific fields.

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