Uniform/Exponential PDF

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In summary, to find the PDF of Z, we can use the transformation method by first finding the CDF of Z and then differentiating it to get the PDF.
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Kalinka35
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Homework Statement


Let X be uniformly distributed over [0, 1] and Y be exponentially distributed with λ=1. Assuming X and Y are independent, give the PDF of Z=X/Y.


Homework Equations





The Attempt at a Solution


I know the individual PDFs of X and Y. I was trying to use a geometric approach to find the CDF and then integrate but since Y is exponential the geometry doesn't really work. Are there any other methods I can use to get this? I haven't seen any examples resembling this.

Thanks.
 
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To find the PDF of Z, we can use the transformation method. First, we need to find the CDF of Z, which can be done by using the following formula:

FZ(z) = P(Z ≤ z) = P(X/Y ≤ z) = P(X ≤ zY)

Since X and Y are independent, we can rewrite this as:

FZ(z) = P(X ≤ z)P(Y ≤ Y) = (z)(1-e^(-λY))

Next, we can differentiate the CDF to get the PDF of Z:

fZ(z) = d/dz FZ(z) = 1-e^(-λY)

Substituting in the value of Y as 1 (since λ=1), we get:

fZ(z) = 1-e^(-1)

Therefore, the PDF of Z is 1-e^(-1). I hope this helps! Let me know if you have any further questions or if you need clarification on any steps.
 

What is a Uniform PDF?

A Uniform PDF (Probability Density Function) is a statistical distribution that describes the likelihood of a continuous random variable falling within a given range. It is characterized by a constant probability for all values within the range, resulting in a rectangular shape when graphed.

What is an Exponential PDF?

An Exponential PDF is a statistical distribution that describes the likelihood of a continuous random variable following an exponential decay. It is characterized by a high probability for smaller values that decreases exponentially as the value increases, resulting in a right-skewed curve when graphed.

What are the main differences between Uniform and Exponential PDFs?

The main difference between Uniform and Exponential PDFs is their shape. A Uniform PDF has a constant probability for all values within a given range, resulting in a rectangular shape when graphed. An Exponential PDF has a high probability for smaller values that decreases exponentially as the value increases, resulting in a right-skewed curve when graphed. Additionally, Uniform PDFs are used for describing a random variable with equal likelihood for all values, while Exponential PDFs are used for modeling the time between events in a process that follows an exponential decay.

What are some real-world applications of Uniform and Exponential PDFs?

Uniform PDFs can be used to describe the distribution of continuous random variables in fields such as finance, engineering, and physics. For example, the distribution of stock prices or the height of adult humans can be approximated using a Uniform PDF. Exponential PDFs are commonly used to model the time between events in natural processes, such as radioactive decay, population growth, and equipment failure. They are also used in fields such as queuing theory and reliability engineering.

How are Uniform and Exponential PDFs related to other probability distributions?

Uniform and Exponential PDFs are both types of continuous probability distributions. They are related to other distributions such as the Normal, Poisson, and Binomial distributions through various transformations and limiting cases. For example, when a Uniform PDF is scaled and shifted, it can approximate a Normal distribution. When the exponential rate parameter is large, an Exponential PDF can approximate a Poisson distribution. Understanding the relationships between probability distributions is important in applied statistics and data analysis.

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