Chapter 3: Problem 13
Using the computer, obtain plots of beta pdfs for \(\alpha=1,5,10\) and \(\beta=\) \(1,2,5,10,20 .\)
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Consider a multinomial trial with outcomes \(1,2, \ldots, k\) and respective probabilities \(p_{1}, p_{2}, \ldots, p_{k} .\) Let ps denote the \(\mathrm{R}\) vector for \(\left(p_{1}, p_{2}, \ldots, p_{k}\right) .\) Then a single random trial of this multinomial is computed with the command multitrial (ps), where the required \(\mathrm{R}\) functions are: \({ }^{2}\) (a) Compute 10 random trials if \(\mathrm{ps}=\mathrm{c}(.3, .2, .2, .2, .1)\). (b) Compute 10,000 random trials for ps as in (a). Check to see how close the estimates of \(p_{i}\) are with \(p_{i}\).
For this exercise, the reader must have access to a statistical package that obtains the binomial distribution. Hints are given for \(\mathrm{R}\) code, but other packages can be used too. (a) Obtain the plot of the pmf for the \(b(15,0.2)\) distribution. Using \(\mathrm{R}\), the following commands return the plot: \(x<-0: 15 ;\) plot \(\left(\operatorname{dbinom}(x, 15, .2)^{-} x\right)\) (b) Repeat part (a) for the binomial distributions with \(n=15\) and with \(p=\) \(0.10,0.20, \ldots, 0.90 .\) Comment on the shapes of the pmf's as \(p\) increases. Use the following \(\mathrm{R}\) segment: \(\mathrm{x}<-0: 15 ; \quad\) par \((\mathrm{mfrow}=\mathrm{c}(3,3)) ; \mathrm{p}<-1: 9 / 10\) for \((j\) in \(p)\left\\{\right.\) plot \(\left(\right.\) dbinom \(\left.(x, 15, j)^{\sim} x\right) ;\) title(paste \(\left.\left.(" p=", j)\right)\right\\}\) (c) Let \(Y=\frac{X}{n}\), where \(X\) has a \(b(n, 0.05)\) distribution. Obtain the plots of the pmfs of \(Y\) for \(n=10,20,50,200 .\) Comment on the plots (what do the plots seem to be converging to as \(n\) gets large? ).
Let \(X_{1}\) and \(X_{2}\) be independent with normal distributions \(N(6,1)\) and \(N(7,1)\), respectively. Find \(P\left(X_{1}>X_{2}\right)\). Hint: \(\quad\) Write \(P\left(X_{1}>X_{2}\right)=P\left(X_{1}-X_{2}>0\right)\) and determine the distribution of \(X_{1}-X_{2}\)
Let \(X\) have a geometric distribution. Show that $$ P(X \geq k+j \mid X \geq k)=P(X \geq j) $$ where \(k\) and \(j\) are nonnegative integers. Note that we sometimes say in this situation that \(X\) is memoryless.
Determine the constant \(c\) so that \(f(x)=c x(3-x)^{4}, 0
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