alt="left-parenthesis upper X comma upper Y right-parenthesis"/> 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.
