Let $X_1$ denote the weight (in tons) of a bulk item stocked by a supplier at the beginning of a week and suppose that $X_1$ has a uniform distribution over the interval $0 \le X_1 \le 1$. Let $X_2$ denote the amount (by weight) of this item sold by the supplier during the week. Given the value $X_1=x_1$, $X_2$ has a uniform distribution over the interval $[0,x_1]$, where $x_1$ is a specific value of $X_1$.
a)Find the joint density function of $X_1$ and $X_2$.
b.If the supplier stocks a half-ton of the item, what is the probability that she sells more than a quarter ton?
c. If it is known that the supplier sold a quarter ton of the item, what is the probability that she had stocked more than a half-ton?
Since the two variables are not indepent, I find it hard to say anyting about their joint density function. What I first of all did was find the PDF for $X_1$ which is obviously just 1. For $X_2$ the PDF should be $1/x_1$, I would say. However, how do these help me to find the joint density function?
Best Answer
Yes you're right for c), you need to calculate $$ P(X_1 > \frac 1 2 \vert X_2 = \frac 1 4). $$ Ok, you know that $X_2 \vert X_1$ is uniformly distributed on $(0,X_1)$ so $$ P(X_2 > \frac 1 4 \vert X_1 = \frac 1 2) = P( \mathcal U(0,\frac 1 2) > \frac 1 4) = \int_{1/4}^{1/2} 2 dx = \frac 1 2. $$ For c), calculate the density of $X_2 \vert X_1$ using Bayes formula and do the same.