# Beta random Variable

I have some conceptual hiccups ( for a lack of better word ) with beta Random variables.

Here's the question:

The number of students who get failing grade in a hard test is given by Beta B(2,3) random variable.

α = 2, β= 3

To determine the number of students that fail, I found the Mode using the formula Mode = (α−1)/(α+β−2)

I get (2-1)/(2+3-2) = 1/3

Therefore, the number of students that fail is 1/3rd.

Is this right?

Now how do I find the the probability of 80% students passing the test?

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No. You can't "determine the number that fail": that's why "random" is part of "random variable". Also, presumably this should be the "fraction of students" (something between 0 and 1), rather than the "number of students" which would be an integer. The mode is indeed $1/3$, but that's just the value where the density is largest; the mean is $2/5$ and the median is approximately $.38572756813239$; the actual value could be anywhere from 0 to 1. To get the probability that 80% pass, you'll want to integrate the density from $0.20$ to $1$.
From $0$ to $0.2$, actually. – Did Oct 31 '11 at 8:20