# hw-3

Homework 3 in STAT400: Computational Statistics @ CSU

## Assignment

Let \(X \sim \text{Uniform}(a, b)\). Derive \(E[X]\) and \(Var[X]\).

- Let \(X \sim \text{Exponential}(\alpha)\).
- Derive \(E[X]\) (show the parts we skipped in class).
- What are the parameter(s) of the exponential distribution and what values can they take?

- Let \(X\) = the outcome when a fair die is rolled.
- Find \(E[X]\)
- Find \(Var[X]\)
- Before the die is rolled you are offered either \(1/3.5 = \$0.29\) (guaranteed amount) or \(h(X) = 1/X\) dollars (random amount). Would you accept the guaranteed amount or would you gamble? In your answer, discuss what this means about \(1/E[X]\) as compared to \(E[1/X]\). In particular, does \(1/E[X]\) always equal \(E[1/X]\)?

Give three examples of Bernoulli random variables.

- For each of the random variables defined below, describe the sample space (set of possible values) and state whether the random variable is continuous or discrete.
- \(X=\) the number of unbroken eggs in a dozen eggs.
- \(Y=\) the pH of a randomly chosen soil sample.
- \(Z=\) the number of CSU students who skipped their first class.
- \(W=\) the distance between CSU and the local residence of a randomly chosen CSU student.

- Let \(X=\) the number of days of sick leave taken by a randomly selected employee of a large company during a particular year. The maximum allowable days per year is \(14\). Let the following values of the cdf be defined \[
F(0) = 0.58, F(1) = 0.72, F(2) = 0.76, F(4) = 0.88, F(5) = 0.94.
\]
- What is the sample space of \(X\)?
- Compute \(P(2 \le X \le 5)\)
- Compute \(P(X \ge 5)\) and \(P(X > 5)\).

- The cdf for a random variable \(X\) is given below. \[
F_X(x) = \begin{cases}
0 & x < -2 \\
\frac{1}{2} + \frac{3}{32}(4x - x^3/3) & -2 \le x < 2 \\
1 & x \ge 2
\end{cases}
\]
- Compute \(P(X > 0.5)\)
- Find \(f_X(x)\).
- The median \(\tilde{\mu}\) of a continuous random variable is the \(50^{th}\) percentile of the distribution, given by \(0.5 = F_X(\tilde{\mu})\). Show that \(\tilde{\mu} = 0\).

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