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A graduate program has full-time and part-time students. The number of full-time students who join the program each year is a Poisson random variable with a mean of $30$. The number of part-time students who join the program each year is a Poisson random variable with a mean of $20$. $20\%$ of full-time students graduate in $1.5$ years and $80\%$ graduate in $2$ years. $40\%$ of part-time students graduate in $3$ years and $60\%$ graduate in $4$ years. On average, at a given moment in time, what fraction of enrolled students are full-time and what fraction are part-time?

My thought: First, the average number of years a full-time student graduate $= 0.2*1.5+0.8*2 = 1.9$ years, and average number of years a part-time student graduate $ = 0.4*3+0.6*4 = 3.6$ years. By Little's Law, if we define $L =$ average number of students enrolled, $\lambda =$ average number of students who join the program each year, and $W = $ average number of years in school till graduate (so either $1.9$ years for full-time and $3.6$ years for part-time). So $L = 30*1.9 + 20*3.6 = 129$ students.

But I'm stuck on how to proceed further using this $L$ to find the fraction of enrolled students on average that are full-time. Could anyone please give some help here? I would really appreciate it.

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I think you are almost done using the following observation. If $X_{1}$ and $X_{2}$ are Poisson with $\lambda_{1}$ and $\lambda_{2}$ respectively, then distribution of $X_{1}$ conditional on $X_{1}+X_{2}=k$ is binomial with $n=k$ and $p=\frac{\lambda_{1}}{\lambda_{1}+\lambda_{2}}$. See more here.

I think this makes the distribution of full-time students binomial with $n=129$ and $p=\frac{3}{5}$. Mean of this is $np$. Or maybe the $\lambda$s should be scaled to factor in the fact that full- and part-time students exit at different rates.

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    thank you for your help. But the question asks for fraction, and based on your work above, the answer would be $p=\frac{3}{5}$ for full-time students and $\frac{2}{5}$ for part-time students. This means it is completely independent of the exit rates. I think this might be true, although I have not figured out why... – 2017-01-27
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    I think you are correct that "$\lambda$s should be scaled to factor." The reason is because $X_1+X_2$ only accounts for the number of students JOINING in the program each year, NOT enrolling. But is it correct that we should scale $\lambda_1$ by $1-\frac{1}{1.85}$, and the other $\lambda_2$ by $1-\frac{1}{3.5}$? – 2017-01-27
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    my bad. the fraction should be $\frac{1}{1.9}$ and $\frac{1}{3.6}$. This is the FRACTION of a number of full-time and part-time students graduating per year, respectively. – 2017-01-27