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.
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.