Robert P. Dobrow - Probability

8 Chapter 7FIGURE 7.1: Three normal distributions.FIGURE 7.2: Normal approximation of the binomial distribution. For fixed , ...FIGURE 7.3: Approximating the binomial probability using a normal distribu...FIGURE 7.4: Four gamma distributions.FIGURE 7.5: The histogram is from simulating the sum of 20 exponential varia...FIGURE 7.6: Poisson process: relationships of underlying random variables.FIGURE 7.7: Four beta distributions.FIGURE 7.8: Scale-invariance of the Pareto distribution. The density curve i...

9 Chapter 8FIGURE 8.1: Simulated distribution of , where The curve is the density fu...FIGURE 8.2: The geometry of the lighthouse problem.FIGURE 8.3: Simulating from density using the inverse transform method.FIGURE 8.4: Simulated distribution of three independent exponential random v...FIGURE 8.5: Distributions from (a) to (c): uniform, sum of two independent u...FIGURE 8.6: The shaded region is the event that Angel and Lisa meet.FIGURE 8.7: Let be uniformly distributed on the triangle. The shaded regio...FIGURE 8.8: Region for and .FIGURE 8.9: Geometry of Buffon's needle problem. The needle intersects a lin...FIGURE 8.10: Buffon's needle problem is solved by finding the area under the...

10 Chapter 9FIGURE 9.1: (a) Graph of a joint density function. (b) Conditional density o...FIGURE 9.2: Graph of conditional density of given . The density function ...FIGURE 9.3: Simulations of (a) joint distribution of , (b) marginal distrib...FIGURE 9.4: Cumulative distribution function of insurance payout has jump di...FIGURE 9.5: Number of exams versus number of A's. Correlation is 0.775.FIGURE 9.6: Distribution of arrowtooth flounder in Alaskan fisheries. Ellips...FIGURE 9.7: Contour plots of standard bivariate normal densities with (a) ,...FIGURE 9.8: Bivariate standard normal distribution.FIGURE 9.9: Galton's height data for fathers and sons are well fit by a biva...FIGURE 9.10: Plot of 1000 observations from bivariate standard normal distri...

11 Chapter 10FIGURE 10.1: Mathematician John Kerrich tossed 10,000 coins when he was inte...FIGURE 10.2: An illustration of the weak law of large numbers for Bernoulli ...FIGURE 10.3: Four realizations of convergence of in 1000 fair coin flips....FIGURE 10.4: Sequences of averages for the Cauchy distribution whose expec...FIGURE 10.5: Using probability and random numbers to estimate the area of th...FIGURE 10.6: Histogram of from an underlying sequence of Bernoulli trial...FIGURE 10.7: The simulated distribution of for four population distributio...FIGURE 10.8: Four random walk paths of 10,000 steps. The horizontal axis rep...FIGURE 10.9: The normalized sum of just six dice throws comes close to the n...

12 Chapter 11FIGURE 11.1: Graph on four vertices.FIGURE 11.2: (a) Cycle graph on vertices. (b) Complete graph on vertices...FIGURE 11.3: The -hypercube graph for (a) and (b).FIGURE 11.4: Weighted graph.FIGURE 11.5: Weighted graph for the general two-state Markov chain.FIGURE 11.6: Gibbs sampler simulation of bivariate standard normal with .FIGURE 11.7: Distribution of the marginal distribution of from 10,000 runs...FIGURE 11.8: Lollipop graph on nine vertices.FIGURE 11.9: Weighted graph.