# Generating three random numbers from normal distribution using 3 independent uniform numbers

Each of the variables A,B,C follows a normal distribution with E(A)=E(B)=E(C)=1,V(A)=1,V(B)=2,V(C)=4. Correlation between A-B=0.2,B-C=0,AC=0.8. We are given three independent uniform (on[0,1]) random observations 0.426,0.238,0.927. From these three random numbers, generate a set of values of A,B,C justifying your method.

We just have started learning simulation and generating random numbers. This question completely flew over my head. How to involve correlation and generate 3 random normal numbers from 3 independent uniform.Is there any way, without using the error function?

• See stats.stackexchange.com/search?q=Box+Muller for one efficient simple method. There are many other methods that do not require transcendental functions like the error function, because many other distributions (including Gamma, Beta, Chi-squared, F-ratio) have straightforward relationships with the Normal distribution.
– whuber
Apr 24 '19 at 14:36

Using the decomposition $$f_{A,B,C}(a,b,c)=f_{A}(a)f_{B|A}(b|a)f_{C|A,B}(c|a,b)$$ of the joint Normal density into one Normal marginal density and two Normal conditional densities leads to the representation $$A=F_A^{-1}(U_1)\quad B=F_{B|A}^{-1}(U_2|A)\quad C=F_{C|A,B}^{-1}(U_3|A,B)$$ where the three inverse cfs (or quantile functions) are location scale transforms of the standard Normal inverse cdf (see, eg, qnorm in R).
• Do you agree with the decomposition of the joint into three progressively conditional densities? Simulating $(A,B,C)$ means simulating $A$, then $B$ conditional on the simulated $A$, then $C$ conditional on the simulated $A$ and $B$. Apr 29 '19 at 11:23