Questions

Birthdays

Use Poisson approximations to investigate the following types of coincidences. The usual assumptions of the birthday problem apply, such as that there are 365 days in a year, with all days equally likely.

Q

How many people are needed to have a 50% chance that at least one of them has the same birthday as you?

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Q

How many people are needed to have a 50% chance that there are two people who not only were born on the same day, but also were born at the same hour (e.g., two people born between 2 pm and 3 pm are considered to have been born at the same hour).

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Uniform

Let $U \sim Unif(-1,1).$ Note that the PDF of $U$ is $f(x) = \frac{1}{2}$.

Q

Compute $E(U)$

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Q

Compute $Var(U)$

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Q

Compute $E(U^4)$

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Q

Find the CDF of $U^2$.

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Q

Find the PDF of $U^2$.

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Q

Is $U^2$ a uniform distribution?

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Normal

Let $Z \sim N(0,1)$ with CDF $\Phi$. The PDF of $Z^2$ is the function given by:

$$g(w) = \frac{1}{\sqrt{2\pi w}} e^{-w/2}$$

with a support of $w \geq 0$.

Q

Find expressions for $E(Z^4)$ as integrals in two different ways one based on the PDF of $Z$ and the other based on the PDF of $Z^2$.

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Q

Find $E(Z^2 + Z + \Phi(Z))$.

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Q

Find the CDF of $Z^2$ in terms of $\Phi$; do not find the PDF of $g$.

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