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I am trying to generate samples from a Gaussian distribution, with the sample having a certain probability, but unable to figure out a way. Eg. I want a sample from $N(0, 1)$ which has a probability of $0.3$.

For context, I am trying approximate inference for a Bayesian network.

For discrete variables, I divide the real line between 0 and 1 at the intervals equal to the value of probability for each of the discrete value. $P(X=x_0)=0.35$, $P(X=x_1) =0.65$, using random number generator, if the generated number is $\le 0.35$, $X$ is instantiated to $x_0$ else to $x_1$.

Unable to figure out a way to do the same for continuous distribution.

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closed as unclear what you're asking by Juho Kokkala, Christoph Hanck, Silverfish, Nick Cox, kjetil b halvorsen Mar 18 '16 at 10:18

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ The meaning of 'sample having a certain probability' is unclear. $\endgroup$ – Juho Kokkala Mar 18 '16 at 8:12
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Usually you would use a built-in function of your favourite package (R, numpy etc.) to do this. However, I suggest you first make sure you know what you want to sample from or at least restate your wish to sample from "N(0,1) with probability 0.3". N(0,1) is already a parametrised distribution, like the one you described for the discrete case. If you want to do the sampling manually, you would do so by reversing the process. I.e. draw a number from a uniform distribution and then look for which x the CDF of N(0,1) reaches the particular value you have drawn.

I hope that helps clarifying things a bit.

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With sampling from continuous distributions, like normal, you can follow exactly the same procedure like in your discrete example. First, using cumulative distribution function $F_X$ you find probability that $X \le x$,

$$ \Pr(X \le x) = F_X(X) $$

and then using inverse of cumulative distribution function (so called, quantile function) you follow the procedure described by you. So first you sample $U$ that is uniformly distributed in $[0,1]$ and then take

$$ X = F_X^{-1}(U) $$

in your case it is quantile function of standard normal

$$ X = \Phi^{-1}(U) $$

or in R code:

U <- runif(1000)
X <- qnorm(U)

See this thread for more details: How does the inverse transform method work?

There are also other algorithms for sampling from normal distribution, as described, for example, in Wikipedia.

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