I see that you've uploaded an image. I'll extract the details and solve Question 1 for you. Let me review the content of the image. PIL import Image

Open the image to inspect the contents

image_path = '/mnt/data/1733721927606wjrt2xus.jpg' img = Image.open(image_path) img.show()I've displayed the image, but it appears I need to analyze the content further to extract the mathematical problem. I'll proceed with extracting the text now. pytesseract

Extract text from the image using pytesseract

extracted_text = pytesseract.image_to_string(img) extracted_textFrom the extracted text, it appears that Question 1 involves statistical distributions. The question specifically asks:

• "If X and Y are independent random variables, find the probability density function (P.D.F.) for the product XY and identify its distribution."

Step-by-step Solution:

We need to find the probability density function (P.D.F.) of the product of two independent random variables $X$ and $Y$, and identify its distribution. Let’s break down the process:

1. Given Information:

   • $X$ and $Y$ are independent random variables.
   • We need to find the P.D.F. for $Z = XY$, where $Z$ is the product of $X$ and $Y$.

2. Methodology:

   • To find the distribution of the product of two independent random variables, we use a standard method from probability theory. The P.D.F. of the product $Z = XY$ is given by the convolution integral: $f_Z(z) = \int_{-\infty}^{\infty} \frac{f_X(x) f_Y(z/x)}{|x|} dx$ where:
     • $f_X(x)$ is the probability density function of $X$,
     • $f_Y(y)$ is the probability density function of $Y$,
     • and $|x|$ in the denominator is there to account for the change of variable when transforming to the product.

3. Conclusion:

   • To identify the distribution, we would need to know the specific distributions of $X$ and $Y$ (e.g., normal, exponential, etc.).
   • If $X$ and $Y$ are of certain known distributions, such as two independent exponential distributions, we could further deduce the exact form of the distribution for $Z$.

However, for a complete answer, additional information about the specific distributions of $X$ and $Y$ is required. Would you be able to provide more context on their distributions?

If you'd like to proceed further, feel free to ask for details on a specific example or additional questions.

Here are some related questions to explore:

1. What is the moment generating function (MGF) of the product of independent random variables?
2. How do you find the distribution of a sum of independent random variables?
3. What is the convolution formula used for finding the P.D.F. of sums and products?
4. How can the product of two independent Gaussian distributions be modeled?
5. What are some practical applications of understanding the distribution of the product of random variables?

Tip: Always verify the independence of variables before using the product formula for their joint distribution.