Uniformly Distributed ingredients Suppose we need to make a dish that has three ingredients A, B and C. All are distributed uniformly between [0, 2], [0, 2], [0, 1] respectively.
To create the dish, we need 1/4 of A, 1/4 of B and 1/8 of C.
How can we express number of dishes we can make, say X w.r.t A, B and C (X can be non integer). What is the CDF of X?
I am stuck on this problem from the homework. I know that we can use the convolution to take sum of A, B and C but not sure how to observe the individual requirements.
Any help will be highly appreciated. Thanks.
 A: Weird problem! Maybe one should not make up probability problems when one is hungry. The random variable $X$ (the "number" of dishes we can make) is allowed to be a real number. Let $F(x)$ be the cdf of $X$. 
Note that $X\ge x$ if we have at least $x/4$ of ingredient A, and at least $x/4$ of B, and at least $x/8$ of C. For $0\le x\le 8$, we have the following probabilities:
The probability we have at least $x/4$ of ingredient A is $1-(x/4)/2=1-x/8$. 
The probability that we have at least $x/4$ of B is also $1-x/8$. The probability we have at least $x/8$ of C is also $1-x/8$. 
Now we have to assume independence of the amounts of A, B, C available. Then for $0\le x\le 8$, the probability that $X\ge x$ is $(1-x/8)^3$. 
It follows that $F(x)=\Pr(X\le x)=1-(1-x/8)^3$ for $0\le x\le 8$. 
For completeness, note that $F(x)=0$ if $x\lt 0$, and $F(x)=1$ if $x\gt 8$.
Remark: A similar analysis could be made if the number of dishes is supposed to be an integer. Writing out the details is somewhat messier.
