# Finding expression

I don't understand this at all..

We have a Poisson distribution with parameter μ. The probability to pass the test (for the students) is p=2/3. Let X be the amount of students to pass the test. Find an expression for P(X=k). Which probability distribution does X have?

This is more or less my homework problem. Finding the expression for P(X=k). What does it mean? Am I wrong if I think that the X is an uniform distribution? Because for all students that pass the test, the probability is the same?

Thanks

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To my knowledge, finding the expression for $P(X=k)$ means to find an expression for the probability that the number of students that pass the class is equal to some (arbitrary) constant value. – apnorton Mar 7 '13 at 2:51
Let Y denote the number of students who try the test. Apparently Y is Poisson with parameter µ. Can you compute the distribution of X conditionally on Y? And deduce the distribution of X? – Did Mar 7 '13 at 3:13
The random variable $X$ does not have uniform distribution. After some work, or perhaps intuition, you can find that it has Poisson distribution, with a parameter different from $\mu$. – André Nicolas Mar 7 '13 at 3:26
But, the chance to pass the test is for all students the same? – Asa1789 Mar 7 '13 at 3:33
OP How does the accepted answer answer the question Find(ing) the expression for P(X=k)? – Did Mar 7 '13 at 8:05

The classical example for Poisson distribution is the number of spelling mistake in Pages of a book. In the same notion, here $X$ is a poisson random variable with some mean $\mu$ and the expression $P(X=k)$ is the probability of the exact $k$ student passed in the examination.