ECE 650. Some MATLAB Help (addendum to Lecture 1) D. Van Alphen (Thanks to Prof. Katz for the Histogram PDF Notes!)

Transcription

1 ECE 65 Some MATLAB Help (addendum to Lecture 1) D. Van Alphen (Thanks to Prof. Katz for the Histogram PDF Notes!)

2 Obtaining Probabilities for Gaussian RV s - An Example Let X be N(1, s 2 = 4). Find Pr(X < 3). Pr( X 3) Pr Z 3 m Pr Z s (1) For 2 nd approach below Standard Normal PDF MATLAB Code* (2 approaches): >> F1 = normcdf(3, 1, 2) F1 =.8413 >> F2 = normcdf(1,, 1) F2 =.8413 MATLAB function normcdf(x, m, s) returns F X (x) for RV X ~ N(m, s 2 ) ECE 65 D. van Alphen 2

3 Obtaining Probabilities for Gaussian RV s - Another Example Let Z be N(, 1). Find Pr(-1 < Z < 1) = P( Z < 1). Pr(-1 < Z < 1) = (1) (-1) MATLAB Code: >> diff = normcdf(1,, 1) normcdf(-1,,1).5.4 (-1) (1) diff = Probability within 1 standard deviation of the mean ECE 65 D. van Alphen 3

4 Obtaining Probabilities for the Standard Normal Random Variable, Z: N(, 1) Probability within 1 s of the mean Probability within 2 s of the mean Probability within 3 s of the mean Pr( Z < 1) 68% Pr( Z < 2) 95.4% Pr( Z < 3) 99.7% % 95.4% 99.7% ECE 65 D. van Alphen 4

5 pdf (By Claims 1 & 2, Lecture 1, p. 36): For Any RV X ~ N(m, s 2 ) -1 s +1 s 68% -2 s +2 s 95.4%.5-3 s +3 s 99.7% Pr( X m < 1s ) = P( Z < 1) 68% Pr( X m < 2s ) = P( Z < 2) 95.4% Pr( X m < 3s ) = P( Z < 3) 99.7% m ECE 65 D. van Alphen 5

6 MATLAB Functions for Gaussian RV s normcdf(x, m, s) computes the cdf, F(x) = Pr(X x) for RV X ~ N(m, s 2 ) norminv(p, m, s) computes the inverse cdf for RV X ~ N(m, s 2 ); i.e., it returns the value x such that F X (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 65 D. van Alphen 6

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 1-bin histogram of measured voltages for 1 nominal 1-KW resistors MATLAB Command: >> hist(y) data vector Resistance, KW ECE 65 D. van Alphen 7

8 Histograms, continued MATLAB allows you to specify the number of bins: >> hist(y, 2) (a 2-bin histogram of data vector y) MATLAB allows you to specify the bin centers: Resistance, KW >> centers = [7 : 15]; >> hist(y, centers) ECE 65 D. van Alphen 8 Resistance, KW

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) % histogram >> rel_freq = freq/sum(freq); >> bar(centers, rel_freq, 1) % rel. freq. histogram >> dens = rel_freq/.5; % bin width =.5 >> bar(centers, dens, 1) % density histogram Note: The 1 in the 3 rd argument of bar makes the bars adjacent, without spaces between them.) ECE 65 D. van Alphen 9

10 Frequency Histogram Rel. Freq. Histogram Density Histogram 25 Histogram.25 Rel. Freq. Histogram Resistance, KW.5 Density Resistance, KW Total area = 1, like pdf Resistance, KW ECE 65 D. van Alphen 1

11 Density Histograms and PDF s Recall that the density histogramgives the distribution of relative frequencies divided by the bin width for the data Density Histogram # data pts bin width Probability Density Function Generating 1, variates from the distribution: N(, 1) >> x = normrnd(, 1, 1, 1); To generate the theoretical pdf: >> x1 = -4 :.1 : 4; >> y1 = normpdf(x1,, 1); >> plot(x1, y1) ECE 65 D. van Alphen 11

12 .4.3 Density Histogram PDF 1, Observations of a Standard Normal RV bins 3 bins bins pdf ECE 65 D. van Alphen 12

13 MATLAB Function Summary (Requiring the Statistics Toolbox) (compare to p. 6) 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 F X (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 Note: On p. 6, the prefix dist is replaced by the prefix norm, for normal RV s. ECE 65 D. van Alphen 13

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 or Normal (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 65 D. van Alphen 14

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 65 D. van Alphen 15

16 Some MATLAB Practice To find Pr(X < 7) if X is Rayleigh with pdf: x x x x f (x) exp U(x) exp U(x) X s s From: MATLAB: >> doc raylcdf 2 p = raylcdf(x, b) returns the Rayleigh cdf at each value in x using the corresponding scale parameter, b x t 2 t p = F X (x) = exp dt 2 2 b 2b In class: check with graph of pdf. 2 >> raylcdf(7, 2) ans =.9978 D. van Alphen 16

17 More MATLAB Practice Say we transmit 3 bits over a noisy channel, where errors occur independently from bit-to-bit, with probability.1. If RV X is the # of errors appearing in a 3-bit word on reception, find the pdf and cdf for X. Note that the RV is binomial, with N = 3 trials, p =.1 = probability of success. From MATLAB: >> doc binocdf: y = binocdf(x, N, p) computes a binomial cdf at each of the values in x using the corresponding number of trials in N and probability of success for each trial in p. >> x = -1 :.1 : 4; y = binopdf(x, 3,.1); >> z = binocdf(x, 3,.1); stem(x, y), title('pdf') >> figure, plot(x, z), title('cdf'), axis([ ]) D. van Alphen 17

18 .8 pdf cdf D. van Alphen 18