A group of children are playing hide-and-seek. There are five potential hiding s

Question

A group of children are playing hide-and-seek. There are five potential hiding spots and two children hiding. Each hiding spot can only accommodate a single child. The seeker will check hiding spots one-by-one. If a child is hiding in that spot they will be found when it is checked. Let X be the number of spots checked until the first child is found and let Y be the number of additional spots checked until the second child is found. Find the joint PMF of X and Y. Find the marginal PMF for X and the marginal PMF for Y.

Explanation / Answer

Given X be the number of spots checked until the first child found

Y: be the number of additional spots checked until the second child found

here for X and Y we apply geometric distribution before a success there is x failures and similarly before a success there is y failures so the marginal PMFs of X and Y are as follows

f(x) = q^x.p x>0

f(y) = q^y.p y>0

and joint PMF of X and Y is

f(x,y) = q^x.p .q^y.p =q^x+y .p^2

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