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The first part of this question states: Consider a very small town with 50 families with children. Let X be the number of children in a family picked at random from the 50 families with children in the town. Suppose that the family size distribution is given by $ fX(x) = \begin{cases} 0.3 & \text{if } x = 1, \\ 0.4 & \text{if } x = 2, \\ 0.26 & \text{if } x = 3, \\ 0.04 & \text{if } x = 4. \end{cases} $ I then had to calculate the cumulative distribution function and the Expected value and Var(X). I managed to do that all okay. But with the next part of the question I am really stuck: Now suppose that you pick a child at random from the children in this town - each child is equally likely to be picked - and ask the child how many children there are in their family (including the child you asked). Let Y be the size of the child's family. (i) How many children are there in the town?

My thoughts, at first I thought that this might involve me forming a Poisson distribution but they I realised I won't have any parameter to form the distribution with. What distribution would be best to used then? Because it can't be a Bernoulli trial or a Geometric distribution either.

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    Can you figure out how many children of each type of family there are? If so, that should give you the distribution you're looking for. Don't think too complicated; it has nothing to do with geometric distributions or stuff like that; it's a simple question of how many children in families with $k$ children there are if there are $m$ families with $k$ children.2011-04-26

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Hint: Call $N$ the total number of children and $K_x$ the number of families with $x$ children. Write $N$ and $P(X=x)$ as a function of $(K_y)$. Compute the number of children whose family has size $x$. Conclude.

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    Where are the % you do not get? Be as specific as possible. Your first step is to write $N$ the total number of children when you know there are $K_1$ families with $1$ child, $K_2$ families with $2$ children, and so on. So, $N=$... ?2011-04-26