 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2003

Paper 2

For <independent random variables>, their sum has the product of their <probability generating functions>:
$$
G_{X+Y}(z)=\mathbb E[z^Xz^Y]=G_X(z)G_Y(z)=(1-p+pz)^{n+m}.
$$
Comparing polynomial coefficients identifies a <binomial distribution> with parameters $n+m,p$:
$$
\boxed{\mathbb P(X+Y=k)=\binom{n+m}{k}p^k(1-p)^{n+m-k},\qquad k=0,\ldots,n+m,}
$$
and the <probability> is zero at all other values. The common success <probability> is essential to this binomial form; independent binomial counts with different <probabilities> do not in general add to a binomial count.

Solution

Past exam of the mathematics course of the University of Cambridge / 2003 / ia / Paper 2 / 3F / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2003 / ia / Paper 2 / 3F / b / Solution

 Ancestors (11)

 Discussion (0) New discussion

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2003 / ia / Paper 2 / 3F / b / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1) New discussion

 Articles by others on the same topic (0)

 Discussion (0) New discussion