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# Questions

## Box-Muller Transformation

Suppose we are told that $$U \sim Unif\left({0, 1}\right)$$ and $$V \sim Expo\left({\frac{1}{2}}\right)$$ independently.

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{% tab title="Q" %}
Find the density function and thus the joint distribution of

$$
\begin{aligned}
X &= \sqrt{V} \sin\left({2 \pi U}\right) \\
Y &= \sqrt{V} \cos\left({2 \pi U}\right)\end{aligned}
$$
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{% tab title="A" %}
To do this, we can obtain the Jacobian and perform a change of variables. Note that it is easier to find the Jacobian of the transformation going for $$\left({U, V}\right) \rightarrow \left({X, Y}\right)$$ since $$X$$ and $$Y$$ are already expressed in terms of $$U$$ and $$V$$. The Jacobian is given as follows.

$$
J = \frac{\partial(x, y)}{\partial(u, v)} = \begin{pmatrix}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \\
\end{pmatrix} = \begin{pmatrix}
\sqrt{v} \cos\left({2 \pi u}\right) \left({2 \pi}\right) & \frac{1}{2 \sqrt{v}} \sin\left({2 \pi u}\right) \\
\sqrt{v} \sin\left({2 \pi u}\right) \left({2 \pi}\right) & \frac{1}{2 \sqrt{v}} \cos\left({2 \pi u}\right) \\
\end{pmatrix}
$$

The Jacobian $$|J|$$ is found to be the determinant of $$J$$ above.

$$
\begin{aligned}
|J| &= \sqrt{v} \cos\left({2 \pi u}\right) \left({2 \pi}\right) \frac{1}{2 \sqrt{v}} \cos(2 \pi u) + \sqrt{v} \sin({2 \pi u}) (2 \pi) \frac{1}{2 \sqrt{v}} \sin ({2 \pi u}) \\
&= ({\cos^2({2 \pi u}) + \sin^2({2 \pi u})}) \pi \\
&= \pi\end{aligned}
$$

Thus, our transformation can be specified as follows. Note that $$X^2 + Y^2 = V$$.

$$
\begin{aligned}
f\_{U, V}(u, v) &= f\_{X, Y}(x, y) |J| \\
f\_{X, Y}(x, y) &= \frac{1}{|J|} f\_{U, V}(u, v) \\
&= \frac{1}{|J|} f\_{U}(u) f\_{V}(v) \\
&= \frac{1}{\pi}\frac{1}{2}\exp\left({- \frac{1}{2}v}\right) \\
&= \frac{1}{2\pi}\exp\left({- \frac{x^2}{2} - \frac{y^2}{2}}\right) \\
&= \frac{1}{\sqrt{2\pi}}\exp\left({-\frac{x^2}{2}}\right)\frac{1}{\sqrt{2\pi}}\exp\left({-\frac{y^2}{2}}\right)\end{aligned}
$$

Hence, because we see that the pdf factors nicely into a term involving $$x$$ and a term involving $$y$$, those terms are in fact the marginal densities of $$X$$ and $$Y$$ (be sure to make sure that the normalizing constants go to the right places). We recognize the marginal densities as standard normal densities. And thus, it follows that $$X$$ and $$Y$$ are i.i.d. $$N(0, 1)$$.
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## Beta, Gamma, and Binomial

Let $$B\sim Beta(\alpha, \beta)$$. Find the distribution of $$1 - B$$ in the following two ways.

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{% tab title="Q" %}
Find the distribution of $$1 - B$$ using a change of variables
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{% tab title="A" %}
Let $$W = 1 - B$$. Then we have $$B = 1 - W$$, and so $$\left| \frac{db}{dw} \right| = |-1| = 1$$. Hence, the PDF of $$W$$ is

$$
f\_W(w) = f\_B(b) \left| \frac{db}{dw} \right| = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}b^{\alpha - 1}(1 - b)^{\beta - 1} = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)}(1 - w)^{\alpha - 1}w^{\beta - 1}
$$

for $$0 < w < 1$$. We recognize this PDF as the $$Beta(\beta, \alpha)$$ distribution, and so $$W \sim Beta(\beta, \alpha)$$.
{% endtab %}
{% endtabs %}

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{% tab title="Q" %}
Find the distribution of $$1 - B$$ using a story proof related to the Gamma distribution.
{% endtab %}

{% tab title="A" %}
Using the bank-post office story, we can represent $$B = \frac{X}{X + Y}$$ with $$X \sim Gamma(\alpha, 1)$$ and $$Y \sim Gamma(\beta, 1)$$ independent. Then $$1 - B = \frac{Y}{X + Y} \sim Beta(\beta, \alpha)$$ by the same story.
{% endtab %}
{% endtabs %}

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{% tab title="Q" %}
How does $$B$$ and $$1-B$$ relate to the Binomial distribution?
{% endtab %}

{% tab title="A" %}
If we use $$Beta(\alpha, \beta)$$ as the prior distribution for the probability $$p$$ of success in a Binomial problem, interpreting $$\alpha$$ as the number of prior successes and $$\beta$$ as the number of prior failures, then $$1 - p$$ is the probability of failure and, interchanging the roles of "success" and "failure," it makes sense to have $$1-p$$ be distributed as the following $$1 - p \sim Beta(\beta, \alpha)$$.
{% endtab %}
{% endtabs %}

## Order Statistics

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{% tab title="Q" %}
Let $$U\_1, \ldots, U\_n$$ be i.i.d. $$Unif(0, 1)$$. Find the unconditional distribution of $$U\_{(n - 1)}$$, and the conditional distribution of $$U\_{(n - 1)}$$ given $$U\_{(n)} = c$$.
{% endtab %}

{% tab title="A" %}
Unconditionally, $$U\_{(n - 1)} \sim Beta(n - 1, 2)$$, using what we know about Uniform order statistics. For the conditional distribution,

$$
\begin{aligned}
P\left({U\_{(n - 1)} \leq x | U\_{(n)} = c}\right) &= P({\textrm{remaining $n - 1$ Uniforms $\leq x$} |\ U\_{(1)} < c, \ldots, U\_{(n - 1)} < c, U\_{(n)} = c)} \\
&= \left({\frac{x}{c}}^{n - 1}\right)\end{aligned}
$$

for $$0 < x < c$$. Hence, the PDF is $$f\_{U\_{(n - 1)} | U\_{(n)}}(x | c) = (n - 1)\left({\frac{x}{c}}^{n - 2} \cdot \frac{1}{c}\right)$$. This is the distribution of $$cX$$ where $$X \sim Beta(n - 1, 1)$$, which can be easily verified with a change of variables. Hence, the conditional distribution of $$U\_{(n - 1)}$$ is that of a scaled Beta!
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{% tab title="Q" %}
Let $$X\sim Bin(n, p)$$ and $$B \sim Beta(j, n - j +1)$$, where n is a positive integer and $$j$$ is a positive integer with $$j \leq n$$. Show using a story about order statistics that $$P(X \geq j) = P(B \leq p)$$ This shows that the CDF of the continuous r.v. B is closely related to the CDF of the discrete r.v. X, and is another connection between the Beta and Binomial.
{% endtab %}

{% tab title="A" %}
Let $$U\_1, \ldots, U\_n$$ be i.i.d. $$Unif(0, 1)$$. Think of these as Bernoulli trials, where $$U\_j$$ is defined to be "successful" if $$U\_j < p$$ (so the probability of success is $$p$$ for each trial). Let $$X$$ be the number of successes. Then $$X \geq j$$ is the same event as $$U\_{(j)} \leq p$$, so $$P(X \geq j) = P(U\_{(j)} \leq p)$$.
{% endtab %}
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