MATH235 Week 7  - Moodle Quiz-assessed problems

Normal score
Let $f(x)=\frac{1}{\sqrt{2\pi \theta }}\exp \left(-\frac{x^2}{2\theta }\right)\text{ where }\theta >0.$ Which is equal to $\frac{d}{d\theta }\log \{f(x)\}$?

1. (a)

   $-\frac{1}{2\theta }+\frac{x^2}{2{\theta }^2}$

2. (b)

   $-\frac{1}{2}\log (2\pi )-\frac{1}{2}\log \theta -\frac{x^2}{2\theta }$

3. (c)

   $-\frac{x^2}{2\theta }\log (\frac{1}{\sqrt{2\pi \theta }})$

4. (d)

   $\frac{1}{2}\log (2\pi \theta )-\frac{x^2}{2\theta }$

5. (e)

   $-\log (2\pi \theta )-\frac{x^2}{2\theta }$

[marks: 2]

Normal pdf
The $f(x)$ in Q2 represents a certain normal distribution. Which of the following is the only correct statement about it?

1. (a)

   $\mu =\theta$, ${\sigma }^2=1$

2. (b)

   $\mu ={\sigma }^2=\theta$

3. (c)

   $\mu =0$, ${\sigma }^2=\theta$

4. (d)

   We can say nothing about $\mu$, and ${\sigma }^2=\theta$

5. (e)

   We can say nothing about either $\mu$ or ${\sigma }^2$

[marks: 2]

Interpreting MLE
Which of these is the correct interpretation of an MLE (maximum likelihood estimate) for $\theta$ in a likelihood $L(\theta )$?

1. (a)

   The MLE is the closest to the true value of $\theta$.

2. (b)

   The MLE is the value of $\theta$ that maximises the probability of the observed data.

3. (c)

   If $L^{\prime }(\widehat{\theta })=0$ then $\widehat{\theta }$ is the MLE of $\theta$.

4. (d)

   The MLE is the value of $\theta$ that generated the observed data.

5. (e)

   The MLE is the value of $\theta$ that we are $95\%$ confident is correct.

[marks: 2]

Continuing with WS7.3, suppose a sample of data $(3,0,2,1,3,2,1,0,2,1)$ is observed.

1. (a)

   The MLE is $5/12$ and ${\theta }_{mom}$ is $1/2$.

2. (b)

   The MLE is $2/3$ and ${\theta }_{mom}$ is $5/12$.

3. (c)

   The MLE is $2/3$ and ${\theta }_{mom}$ is $2/3$.

4. (d)

   The MLE is $1/2$ and ${\theta }_{mom}$ is $2/3$.

5. (e)

   The MLE is $1/2$ and ${\theta }_{mom}$ is $5/12$.

[marks: 2]