Extracting random numbers from a probability distribution Consider the random variable $X$ following a standard normal distribution. Could you clarify what does it mean exactly extracting random numbers from the distribution of $X$?
 A: Say you'd like to choose a random digit from $\{0,1,2,3,4,5,6,7,8,9\}$.
You can split the normal curve into ten segments whose integrals are all equal to $1/10$.The first segment of the distribution of the random variable corresponds to picking the digit $1$, with probability $1/10$. The second segment corresponds to $2$, also with probability $1/10$, and so on.
You've essentially converted a normal distribution into a uniform distribution on the finite set of digits.

Edit: The question isn't what I initially thought it was. You should read about probability density functions, of which the normal curve is a special case. If $f(x)$ is the probability density function of $X$, the important equation is that the probability that $X$ is between $a$ and $b$ is equal to
$$P(a \leq X \leq b)=\int_a^b f(x) \,\mathrm{d}x$$
If the random variable $X$ is normally distributed with mean $\mu$ and variance $\sigma$ then the equation becomes
$$P(a \leq X \leq b) = \frac{1}{\sigma\sqrt{2\pi}}\displaystyle\int_a^b \large{e^{-\large\frac{(x-\mu)^2}{2\sigma^2}}}\,\mathrm{d}x$$
A: If you generate sufficient random numbers from a particular distribution (such as the standard normal), then the shape of the histogram of generated numbers will converge towards the shape of your underlying distribution (the familiar bell shape centered on 0 in the case of the standard normal).
If you have a mechanism available for generating random numbers uniformly on the range 0 to 1 then these numbers can, in principle, be transformed to be random numbers from any given distribution. The cumulative density function (cdf) is a function that maps a variate X to the range 0 to 1. The inverse function maps the range 0 to 1 to the range of the variate. Using the inverse cdf of your chosen distribution to transform random numbers which are uniform on 0 to 1 will result in the transformed values which are random numbers from this chosen distribution.
As noted in the comments to the preceding answer, computer programs generate pseudo-random numbers - that is they use a deterministic process to generate a sequence of numbers which has the appearance of being random in nature.   
