What does drawing sample using Metropolis-Hastings algorithm mean? I am confused with the word "draw samples from any probability distribution P(x)", mean
I apologize for my ignorance, but, drawing sample as i understand, is for example, tossing a coin and writing whether the outcome was a head or a tail, x1 {1,0}, whether a subject was obese, normal or underweight , x2 {1=Obese, 2=Normal, 3=Underweight }, am I wrong ? So please help me understand what "draw samples from any probability distribution P(x)", mean ?
 A: The probability distribution of random variable $X$ is a measure of how probable the occurence of a given realisation of $X$ is, eg. $P(X = x)$ or $P(X \in [a,b])$. Given a probability distribution, you can (nearly) always draw samples from it, which means generating realizations of $X$ that follow this probability distribution.
Statistical software packages include functions to efficiently draw pseudo-random samples from classical probability distributions (bernoulli, uniform, gaussian, gamma, ...). For instance in R executing sample(0:1, 10, replace= TRUE) will generate 10 independent samples of a bernoulli $B(\frac{1}{2})$, which can be interpreted as 10 tosses of a fair coin.
The Metropolis algorithm, among others methods, can be used to draw samples from more complicated probability distributions.
A: Your understanding of drawing a sample was correct. In your example with the coin, you would be drawing samples from a Bernoulli distribution. That is, if you did this repeatedly then the distribution of your observations would converge to the Bernoulli distribution with $p = q = \frac{1}{2}$.
Similarly, drawing samples from an arbitrary distribution means that the distribution of $n$ outcomes in your process would converge to the theoretical distribution as $n \rightarrow \infty$.
Metropolis-Hastings is a computational method for sampling from a distribution. You can see if your sampling procedure is correct by making a histogram of the generated observations and comparing it to your target theoretical probability density function. 
