Exercises Part 4¶

1. Estimating Binomial with CLT, 1/2 correction

Given a Bernoulli Process with $n = 36$ and $p = 0.5$, find $P(S_{n} \leq 21)$. Solution

2. Sample Variance for Normal Distribution

The time it takes a central processing unit to process a certain type of job is normally distributed with mean 20 seconds and standard deviation 3 seconds. If a sample of 15 such jobs is observed, what is the probability that the sample variance will exceed 12 ? Solution

3. MLE Estimate

Suppose we observe $n$ independent and identically distributed samples $x_{1}, x_{2}, \ldots, x_{n}$ from an exponential distribution. Estimate the parameter of the exponential. Solution

4. CRLB for an exponential

Let $X_{1}, \ldots, X_{n}$ be a random sample from an eponential distribution with the probability density

\[ \begin{aligned} f_{X}(x;\theta) = \begin{cases} \frac{1}{\theta} \exp \big(-\frac{x}{\theta} \big) &\mbox{$x > 0$}\newline 0 &\mbox{otherwise} \end{cases} \end{aligned} \]

where $\theta > 0$. Derive the Cramer-Rao lower bound for the variance of any unbiased estimator of $\theta$. Also prove that $T = \frac{1}{n} \sum_{i=1}^{n} X_{i}$ is the minimum variance unbiased estimator of $\theta$. Solution

5. MLE and Proving Sufficient Statistic

Suppose $X_{1}, X_{2}, \ldots, X_{n}$ are from a distribution with the following distribution

\[ \begin{aligned} f_{X}(x) = \begin{cases} \frac{2x}{\lambda} \exp \big(-\frac{x^{2}}{\lambda} \big) &\mbox{if $x > 0$}\newline 0 &\mbox{otherwise} \end{cases} \end{aligned} \]

Find the MLE estimate of $\lambda$ and show that it is unbiased and a sufficient statistic of $\lambda$. Solution

6. Bayes Estimator for Normal Distribution

Suppose $X_{1}, X_{2}, \ldots, X_{n}$ are from a normal distribution with unknown mean $\theta$ and known variance $\sigma_{0}^{2}$, and suppose the mean has a prior normal ditribution with mean $\mu$ and variance $\sigma^{2}$. Calculate the Bayes estimator for the mean $\theta$. Solution

7. LMS Estimate

Given the prior $f_{\Theta \vert (\theta)}$, uniform in $[4,10]$, and $f_{X \vert \Theta}(x \vert \theta)$ is uniform in $\[\theta-1, \theta+1 \]$, estimate the posterior of $\theta$. Solution

8. Probability Convergence

Let $X$ be uniformly distributed between $[-1,1]$. Let $X_{1}, X_{2},\ldots,X_{n}$ be independently and identically distributed with the same distribution as $X$. Find whether the following sequences are convergent in probability and also find the limit.

1. $X_{i}$

2. $Y_{i} = X_{i}/i$

3. $Z_{i} = (X_{i})^{i}$

Solution

9. Age of Smokers vs Non Smokers

One often hears that the death rate of a person who smokes is, at each age, twice that of a nonsmoker. What does this mean? Does it mean that a nonsmoker has twice the probability of surviving a given number of years as does a smoker of the same age? Solution

10. Simple Hypothesis Test

Let $X_{1}, \ldots, X_{10}$ be a random sample of heights from a $\mathcal{N}(\mu, 20^{2})$ distribution where we want to test the hypothesis $H_{0}: \mu = 30$ vs $H_{a}: \mu \neq 30$. For a significance level of $0.05$ and mean of the $10$ samples as $27$, determine if $H_{0}$ is accepted or not. Solution

11. Most Powerful Test for Variance

Let $X_{1}, \ldots, X_{5}$ be a random sample from a $\mathcal{N}(2, \sigma^{2})$ distribution where $\sigma^{2}$ is unknown. Derive the most powerful test of size $\alpha = 0.05$ for testing $H_{0}: \sigma^{2} = 4$ against $H_{1}: \sigma^{2} = 1$. Solution