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# System prepended metadata

title: Scholarship Allocation

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# Scholarship Allocation

Define variables: 
$x=$ number of $1000 scholarships
$y=$ number of $2500 scholarships
$z=$ number of $4000 scholarships

Note that $1000x+2500y+4000z=500,000$, so we can solve for $z = -\frac{1}{4}x - \frac{5}{8}y + 125$

Next we examine how to calculate the expected values. A capital $E$ is used to denote expected value.

We want to maximize:

E[Graduations] = E[# graduates 3.25-3.5] + E[# graduates 3.5-3.75] + E[# graduates 3.75-4].

To estimate those expected values, you need to know some probabilities (denoted with a capital $P$). We define the following quantities.

* $p_{3.25}=$P(a 3.25-3.5 student graduates without a scholarship)
* $q_{3.25}=$P(a 3.25-3.5 student graduates with a scholarship)
* $p_{3.5}=$P(a 3.5-3.75 student graduates without a scholarship)
* $q_{3.5}=$P(a 3.5-3.75 student graduates with a scholarship)
* $p_{3.75}=$P(a 3.75-4.0 student graduates without a scholarship)
* $q_{3.75}=$P(a 3.75-4.0 student graduates with a scholarship)
* $N_{3.25}=$ # of students with a GPA from 3.25-3.5
* $N_{3.5}=$ # of students with a GPA from 3.5-3.75
* $N_{3.75}=$ # of students with a GPA from 3.75-4.0


E[# graduates 3.25-3.5]= $(N_{3.25}-x)p_{3.25}+xq_{3.25}$
E[# graduates 3.5-3.75]= $(N_{3.5}-y)p_{3.5}+yq_{3.5}$
E[# graduates 3.75-3.5]= $(N_{3.75}-\left(-\frac{1}{4}x - \frac{5}{8}y + 125\right))p_{3.75}+\left(-\frac{1}{4}x - \frac{5}{8}y + 125\right)q_{3.75}$


We need to maximize a 2-variable function after making those 9 assumptions. I created a calculator at https://www.desmos.com/calculator/uywvkgggyy where you can just enter those 9 numbers and it will give you the values you are after. 


