Probability

• Select a random sample, without replacement, of your desired sample size from the numbers 1 to a population size you set. After entering the population size and sample size click on the "sample without replacement" button. The program will then select a random sample to your specifications and you can see all the numbers by hitting reload on your browser and scrolling, or you can save them to a file (on the Mac) by putting the cursor in the large white area, selecting all, and copying.

• You can "look up" a standard normal probability ; just enter the desired z and press the calculate button.

• Plot normal densities . You can see plotted on the same axes a standard normal density in black and a normal density of your choice in blue. Select a standard deviation by clicking along the horizontal line segement at the top; select a mean by clicking along the horizontal axis.

• View real time simulations of a weakly stationary second order process (in black) and, on the same axis, its best mean square error predictor (in blue). The process is fourth order autoregressive and you make the parameter selections. Your selection should have the property that the solutions of the polynomial equation x^4 - c1 X^3 - c2 x^2 - c3 x - c4 = 0 lie inside the unit circle in the complex plane. The choice is not crucial here however. The process in blue is the best prediction process; its value at time n is a function of the values of the process in black at times n-1, n-2, ... See Karlin and Taylor, A First Course in Stochastic Processes, for a derivation of the best predictor.

• You can also simulate the process and see a plot of the values for 200 time points.

• You can simulate a dynamic linear model Y(t) = m(t) + N(t), m(t) = m(t-1) + D(t), where the N's are iid N(0,V) and the D's are iid N(0,W) and see simultaneuosly the optimal (Bayes) predictor of Y(t), in red, given Y(t-1), ..., Y(1).

• You can see realizations of the dynamic linear model and optimal predictions for blocks of 200 trials.

• An M/M/1 queue has Poisson arrivals and exponential service times. You can pick arrival and service rates and watch a real-time simulation. Coded by Johnny Tyme (J. Tong).

• You can simulate Weibull random variables and plot the histogram of your simulation and the density on the same axes. Coded by Russ Keldorph.

• Simulate sample paths of a Poisson process. Coded by Dale Everett.

• Upper tail probabilities and upper cutoffs for chi-squares can be found here. Sorry, the current version does not run on Macs. Coded by Yonatan Feldman.

Statistics

• The tests given on the hypothesis testing sheet are known to perform well and to prove it mathematically takes considerable effort. A good understanding of what makes these tests work can be had with much less effort as in the following example. Suppose a random sample of n = 5 observations is taken from a normal population and we are interested in testing the null hypothesis that the standard deviation is some particular value, which we shall call here the null standard deviation. The appropriate test statistic is
  (n-1)sample variance/null variance
  and if the null hypothesis is true, it follows a chi-square distribution with n-1 = 4 degrees of freedom. However, if some alternative value other than our null value is the correct population standard deviation,then the test statistic does not have a chi-square density. You can select the value of the null and alternative standard deviations and see on the same axes plots of the chi-square density (plotted in black and correct under the null) and the actual density of the test statistic plotted in red ( the correct one under the true alternative value you selected). You will see that as you move the true population standard deviation away from the hypothesized value the test statistic's distribution is concentrated increasingly in one or the other of the two tails of the null distribution. In the next section this behavior is investigated more thoroughly in a different means example through the power function of the test. (If you are using a Mac and Netscape the applet above may not work correctly, however you can use the Mac version. Just click along the axis to select the ratio of variances.)

• In testing equality of two variances, the test statistic no longer has an F distribution. under an alternative hypothesis. You can see the correct probability density of the test statistic, (sample var1)/(sample var2), where the numerator df are 10 and the denominator df are 8, by selecting the actual ratio c = var1/var2 of population variances along the axis provided. Coded by Derrick Whittle.

• In testing statistical hypothesis the probability (called the power of the test ) of rejecting a null hypothesis depends on the true distribution and on the rejection region (decision rule). In this example we investigate the power function of a test of whether the mean of a normal, with known standard deviation 16, is 0 against the alternative that the mean is not 0, based on a single observation. It can be shown that the optimal rejection region whose size is 0.0455 is to reject the null hypothesis if | x | exceeds 32. You can explore the power function of this test as a function of the mean of the true normal distribution; just click the mouse along the axis to select a mean. You will see below the graph the mean you selected and the power (probability of rejecting the null hyppothesis) for your selection. The red curve is a plot of the corresponding normal density and the area of the red portion is the power.

• You can explore p-values for hypothesis tests of a mean based on a random sample of 4 observations from a normal distribution. We know that the tests of the mean of a single normal population which are found on the testing sheet can be recast in terms of p-values to read: reject the null hypothesis at level alpha if the p-value of the data is smaller than alpha. In the applet you can select the alternative hypothesis and find the p-value of the data by the clicking the mouse along the x-axis to set an observed value of the test statistic (a t statistic in this case of course). The black curve is a graph of the t-density with 3 degrees of freedom, which is the correct probability distribution of the test statistic under the null hypothesis, and the area of the shaded portion is the p-value of the particular value of the test statistic which you selected.

• Plotting likelihood functions for i.i.d. samples from a Bernoulli distribution. After you select n, the sample size, and p, the success probability the applet simulates a sample of the required size and plots the log likelihood function upon clicking the mouse while in the window.

• You can run a Monte Carlo estimate of a Poisson parameter. It is the Bayes estimate of the mean q of a Poisson distribution assuming squared error loss and a heirarchical prior of gamma distributions. See the Poisson gamma heirarchy in Lehmann and Casella, Theory of Point Estimation, for details.