ECE 650 Some MATLAB Help (addendum to Lecture 1) D. Van Alphen (Thanks to Prof. Katz for the Histogram  PDF Notes!) MATLAB GUI for Examining PDF’s, CDF’s: >> disttool Choose pdf Select one of 22 types of distributions or cdf Moveable data cursor Set desired parameters, within limits ECE 650 D. van Alphen 2 Obtaining Probabilities for Gaussian RV’s - An Example Let X be N(1, 4). Find Pr(X < 3). 3  m 3  1   Pr( X  3)  Pr  Z    Pr  Z    (1)  2     Standard Normal PDF MATLAB Code* (2 approaches): 0.8 0.6 0.4 0.2 0 -8 -6 -4 -2 0 2 4 6 8 >> x = 3; z = 1; >> F = normcdf(z, 0, 1) F= 0.8413 >> F1 = normcdf(x, 1, 2) F1 = 0.8413 MATLAB function normcdf(x, m, s) returns FX(x) for RV X ~ N(m, 2) ECE 650 D. van Alphen 3 Obtaining Probabilities for Gaussian RV’s - Another Example Let Z be N(0, 1). Find Pr(-1 < Z < 1) = P( |Z| < 1). Pr(-1 < Z < 1) = (1) –  (-1) 0.5 MATLAB Code: (-1) >> z = [-1 1]; >> F = normcdf(z, 0, 1); % 2D vector >> diff = F(2) - F(1) diff = 0.6827  (1) 0.4 0.3 0.2 0.1 Probability within 1 standard deviation of the mean ECE 650 D. van Alphen 0 -2 0 2 4 Obtaining Probabilities for the Standard Normal Random Variable, Z: N(0, 1) Probability within 1  of the mean Probability within 3  of the mean Probability within 2  of the mean Pr( |Z| < 1)  68% Pr( |Z| < 2)  95.4% 0.5 0.5 0.5 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0 ECE 650 0.1 68% -2 0 2 0 0.1 95.4% -2 0 D. van Alphen Pr( |Z| < 3)  99.7% 2 0 99.7% -2 0 2 5 (By Claims 1 & 2, Lecture 1, p. 36): For Any RV X ~ N(m, 2) 0.25 0.2 pdf -1  +1  68% 0.15 -2  0.1 0.05 0 -8 -3  -6 -4 -2 95.4% 99.7% 0 m +2  +3  2 4 6 8 Pr( |X – m| < 1 ) = P( |Z| < 1)  68% Pr( |X – m| < 2 ) = P( |Z| < 2)  95.4% Pr( |X – m| < 3 ) = P( |Z| < 3)  99.7% ECE 650 D. van Alphen 6 MATLAB Functions for Gaussian RV’s normcdf(x, m, s) computes the cdf, F(x) = Pr(X  x) for RV X ~ N(m, s2) norminv(p, m, s) computes the inverse cdf for RV X ~ N(m, s2); i.e., it returns the value x such that FX(x) = Pr(X  x) = p normpdf(x, m, s) computes the pdf, f(x), of the normal distribution with mean m and standard deviation s normrnd(m, s) returns a pseudorandom number (called a variate) from a normal distribution with mean m and standard deviation s. To generate a matrix of size (#_rows, #_cols) of such variates, use: normrnd(m, s, #_rows, #_cols) See randn for a similar function available without the statistics toolbox. ECE 650 D. van Alphen 7 Histograms • A histogram is a bar graph in which the bar heights show the frequency of (random) data occurring over intervals of equal width – The individual bars are referred to as bins • Example: a 10-bin histogram of measured voltages for 1000 nominal 10-KW resistors • MATLAB Command: >> hist(y) 250 200 150 100 50 0 6 data vector ECE 650 8 10 12 14 Resistance, KW D. van Alphen 8 Histograms, continued • MATLAB allows you to specify the number of bins: 120 – >> hist(y, 20) 100 80 60 (a 20-bin histogram of data vector y) 40 20 0 6 8 10 12 14 Resistance, KW • MATLAB allows you to specify the bin centers: 400 – >> centers = [7 : 15]; – >> hist(y, centers) 300 200 100 0 7 8 9 10 11 12 13 14 15 Resistance, KW ECE 650 D. van Alphen 9 Frequency Histogram  Relative Frequency Histogram  Density Histogram • Relative Frequency Histograms show the fraction of values in each bin (freq. hist. heights  # data points) • Density Histograms are scaled so that the total area (in the bins) is equal to one. (rel. freq. hist. heights  bin width) • MATLAB Example (Graphs on next page) >> centers = [7 : .5 : 13]; >> freq = hist(y, centers); >> bar(centers, freq, 1) >> rel_freq = freq/sum(freq); >> bar(centers, rel_freq, 1) >> dens = rel_freq/.5; >> bar(centers, dens, 1) % histogram % rel. freq. histogram % bin width = .5 % density histogram Note: The “1” in the 3rd argument of “bar” makes the bars adjacent, without spaces between them.) ECE 650 D. van Alphen 10 Frequency Histogram  Rel. Freq. Histogram  Density Histogram Histogram 250 0.25 200 0.2 150 0.15 100 0.1  1000 50 0 6 8 10 12 Rel. Freq. Histogram 0.05 0 6 14 Resistance, KW 8 10 12 Resistance, KW 14 Density 0.5 0.4 Total area = 1, like pdf 0.3 0.2  .5 0.1 0 6 8 10 12 14 Resistance, KW ECE 650 D. van Alphen 11 Density Histograms and PDF’s • Recall that the density histogram gives the distribution of “relative frequencies” divided by the bin width for the data # data pts   Probability Density Function Density Histogram bin width  0 Generating 10,000 variates from the distribution: N(0, 1) >> x = normrnd(0, 1, 10000, 1); To generate the theoretical pdf: >> x1 = -4 : .01 : 4; >> y1 = normpdf(x1, 0, 1); >> plot(x1, y1) ECE 650 D. van Alphen 12 Density Histogram  PDF 10,000 Observations of a “Standard Normal” RV 0.4 0.5  10 bins 0.4 0.3  30 bins 0.3 0.2 0.2 0.1 0.1 0 -4 -2 0 2 0 -4 4 0.5 0.5 0.4 100 bins 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 -4 ECE 650  -2 0 2 0 -4 4 D. van Alphen -2 0  2 4 pdf -2 0 2 4 13 MATLAB Function Summary (Requiring the Statistics Toolbox) For Arbitrary RV’s, with dist representing the particular type of RV: • distcdf(x, ... ) computes the cdf, F(x) = Pr(X  x) • distinv(p, ... ) computes the inverse cdf; i.e., it returns the value x such that FX(x) = Pr(X  x) = p • distpdf(x, ... ) computes the pdf, f(x), of the specified distribution defined at the point(s) x • distrnd( ... ) returns a pseudorandom number (called a variate) from the specified distribution ECE 650 D. van Alphen 14 MATLAB Function Summary (Requiring the Statistics Toolbox) • When using the functions from the previous page, for arbitrary RV’s, recall that dist represents the particular type of RV. – Some of the (~22) RV types available: • Gaussian (dist = norm) • Binomial (dist = bino) • Uniform (dist = unif) • Chi-Square (dist = chi2) • Student’s t (dist = t) • Hypergeometric (dist = hyge) • Rayleigh (dist = rayl) • Exponential (dist = exp) • Discrete Uniform (dist = unid) ECE 650 D. van Alphen 15 MATLAB Functions (Requiring the Statistics Toolbox) • disttool brings up a GUI to display pdf’s and cdf’s for a many types of random variables, with settable parameters • randtool brings up a GUI to generate random variates from many (~22) types of random variables, with settable parameters MATLAB Functions (No Toolbox Required) • randn(…) generates random variates from a normal distribution • rand(…) generates random variates from a uniform distribution ECE 650 D. van Alphen 16