Linear function of Gaussian random variable and its distribution– analytical expression guided numerical histogram and the Monte-Carlo computer experiment based histogram 1. The analytical expression for the distribution of a Gaussian distributed RV: 221(),2xXfxexπ−=−∞<<+∞In a simplified notation we can use notation, (0,1)XN∼. 2. MATLAB script – to compare scaled pdf evaluation with histogram M = 1e6; % # of independent trials X = randn(M,1); % N(0,1) random variables dx = 0.1; x = -4:dx:4; % set up a range for x axis hist(X,x); % histogram of the RV X hold on; fx_eval = 1/sqrt(2*pi)*exp(-x.^2/2); % pdf expression evaluated fx_scale = fx_eval*dx*M; % scaling by M*dx to comp. w/ histogram plot(x, fx_scale,'r') a =2; b = 10;
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Probability distribution, probability density function, Histogram, Cumulative distribution function