Does anyone know what is Laplace's theorem and how do you apply Laplace's theorem in probability?

Question

.sam. · Answer

Go on

.sam. · Answer

I have these
Local Laplace's theorem:
$$P_n(m)\approx \frac{1}{\sqrt{npq}}\psi(x)$$
and integral Laplace
$$P_n(m_1\leq m_1 \leq m_2) \approx \Phi (x_2)- \Phi (x_1)$$

anonymous · Answer

i dunno sorry

anonymous · Answer

It's just an approximation to the binomial distribution.  With a sufficient number of trials (n), the binomial distribution becomes essentially a normal distribution with mean np and standard deviation npq.

kainui · Answer

Yeah, and they even put a nice little picture of what @Jemurray3 describes here: http://en.wikipedia.org/wiki/De_Moivre%E2%80%93Laplace_theorem

.sam. · Answer

Does it have any connections with Bernoulli's formula? It says that if the number of trials n>50, Laplace's theorem is used.

anonymous · Answer

Yes, the idea is that if n is sufficiently large, you can pretend that you have a normal distribution instead of a binomial distribution.

.sam. · Answer

Oh I get it now, also what does this mean? "Local Laplace's theorem is used at npq>9"

anonymous · Answer

The normal distribution is continuous, which means that the probability of X being any individual value is equal to zero - you need an interval to calculate probability. The binomial distribution is discrete, though, so it makes sense to ask "What's the probability of exactly 7 successes?" The short answer is just to evaluate the normal distribution as usual, but without resorting to a table. Here: http://www.encyclopediaofmath.org/index.php/Laplace_theorem