I have been stuck for quite a while on this example, and I always attempt to answer any ambiguities on my own. Unfortunately, I've not been able to resolve this one. This is a textbook example, for my own review, since I've taken the class a few years ago. I'm studying Degroot and Shervish's Probability and Statistics, and going over the example 3.9.5 in the 4th edition (p.179). I can't derive the joint pdf that he presented in the example.
Here's the text:
Let $X_1$ be the value of the stocks at the end of one year, and let $X_2$ be the value of the bonds at the end of one year. Suppose that $X_1$ and $X_2$ are independent. Let $X_1$ have the uniform distribution on the interval [1000, 4000], and let $X_2$ have the uniform distribution on the interval [800, 1200]. The sum Y = $X_1$ + $X_2$ is the value at the end of the year of the portfolio consisting of both the stocks and the bonds.
In attempting to come up with the pdf of Y, he did a change of variables, with setting Z = $X_1$, and came up with the following function:
$$f_1(z)f_2(y − z)= \begin {cases} 8.333 × 10^{-7} & \text {for 1000 ≤ z ≤ 4000 and 800 ≤ y − z ≤ 1200,} \\ 0 & \text{Otherwise} \end {cases}$$
My question:
I can't figure out how he got the constant in the function. Since I thought the answer would be:
$$f_1(z)f_2(y − z)= \begin {cases} 1.2 × 10^{-6} & \text {for 1000 ≤ z ≤ 4000 and 800 ≤ y − z ≤ 1200,} \\ 0 & \text{Otherwise} \end {cases}$$
Since $X_1$ and $X_2$ are i.i.d uniformly distributed with the following pdfs:
$$f_1(x_1)= \begin {cases} \frac{1}{3000} & \text {for 1000 ≤ $x_1$ ≤ 4000,} \\ 0 & \text{Otherwise} \end {cases}$$
$$f_2(x_2)= \begin {cases} \frac{1}{400} & \text {for 800 ≤ $x_2$ ≤ 1200,} \\ 0 & \text{Otherwise} \end {cases}$$
Therefore, the function stated in the example is equal to the product of the marginal pdfs, since we have Z = $X_1$ and Y - Z = $X_2$.
So,
$$f_1(z)f_2(y − z) = f_1(X_1) f_2(X_2) $$
What am I missing?
Thank you very much, in advance, for any hint. I apologize if the syntax isn't perfect; this is my first post on the website.
You get the right idea, just multiply the density together.
$$\frac{1}{3000}\cdot \frac1{400}=\frac1{12}\times 10^{-5}=0.08333 \times 10^{-5}=8.333 \times 10^{-7}$$