Interactive Exercises - Chapter 4

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Problem W4.1

Use the widget for the gamma distribution in the scale parameterization (the one used in the course book) for this exercise. Note that the names for the two parameters in the Gamma distribution is not \(\alpha\) and \(\beta\) as in the book, but instead \(\alpha\) and \(\theta\). C’est la vie. 🤷‍♂️

1. Start with the \(\mathrm{Gamma}(3,1)\) distribution and gradually move the first parameter \(\alpha\) toward 1. What happens with the shape of the distribution at \(\alpha = 1\)?

Solution

The distribution for \(\alpha = 1\) becomes highest in the point \(x=0\) with a monotonically decreasing density in \(x\). The Gamma distribution with \(\alpha = 1\) is actually the exponential distribution.

2. Let us explore the effect of the second parameter, the scale parameter \(\theta\).

• Set \(\alpha=2\) and \(\theta=2\). What is the mean and variance? What is \(\mathrm{Pr}(X\leq 3)\)?
• Set \(\alpha=4\) and \(\theta=1\), what is the mean and variance and \(\mathrm{Pr}(X\leq 3)\) now?
• What if \(\alpha=16\) and \(\theta=0.25\)?

Solution

The mean \(E(X)= \alpha \theta\) is the same in all the three settings and the variance \(V(X)=\alpha \theta^2\) decreases as \(\theta\) becomes smaller, and so does \(\mathrm{Pr}(X\leq 3)\).

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Problem W4.2

Use the widget for the beta distribution for this exercise.

1. Start with the \(\mathrm{Beta}(1,1)\) distribution. Is there another name for this distribution?

2. Now, change to \(\mathrm{Beta}(2,2)\), then to \(\mathrm{Beta}(3,3)\), then finally to \(\mathrm{Beta}(10,10)\). What can you say about the shape for these settings?

3. Set \(\alpha\) and \(\beta\) so that most of the density mass is on the right hand side, i.e. for values close to 1. Give a configuration of \(\alpha\) and \(\beta\) achieves this. Finally, set \(\alpha\) and \(\beta\) so that the values close to \(x=1\) has the highest density and the density is monotonically decreasing toward \(x=0\).

4. Can you make the density symmetric around \(x=0.5\) and bathtub shaped with most of the density close to 0 and 1?

   Solution W4.2a.

   The density becomes constant over the support (0,1). This is the uniform distribution on (0,1).

   Solution W4.2b.

   The Beta distribution is symmetric around \(x=0.5\) for all values of \(\alpha\) and \(\beta\) where \(\alpha=\beta\). When the parameters grows larger the density becomes more concentrated around \(x=0.5\).

   Solution W4.2c.

   For example \(\alpha =20\) and \(\beta=2\) gives most of the mass near \(x=1\). Moving \(\beta\) below 1 (still keeping \(\alpha\) at 20) gives a density that has its maximum at \(x=1\) and monotonically decreases as we move down toward \(x=0\).

   Solution W4.2d.

   Setting \(\alpha=\beta\) and both parameters smaller than 1 gives a symmetric bathtub shape.

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