Which are values and not random variables? (Stats)

Question

From a set of random variables $Y=(Y_1,…,Y_n)$ from a known distribution, what expressions are random variables and not fixed numbers?

1. $\bar{Y}=\frac{1}{n}\sum_{i=1}^n {Y_i}$

2. $E(\bar{Y})$

3. $Var(\bar{Y})$

4. $\sum_{i=1}^n ({Y_i-\bar{Y}})^2$

5. $L(\theta ; Y)$ (the likelihood function)

6. The estimator $\hat{\theta}({Y})$

7. $g(\theta ; Y)=\sum_{i=1}^n ({Y_i-\theta})^2$

8. The derivative of $g(\theta ; Y)$ with respect to $\theta$

I am getting my definitions confused from $\text{MLE’s},$ but I think $5)$ is a $\text{RV}$ as it depends on $Y,$ hence $7)$ and $8)$ are. Also $2)$ and $3)$ are random variables too. I think this is correct but can someone verify my answers and explain where I may be going wrong? Thanks!

Answer 1

The only fixed numbers are 2. and 3.

$E(\overline{Y})=\mu$ and $V(\overline{Y})=\frac{\sigma^2}{n}$

You can realize this observing that 1., the sample mean, is a random variable while 2. and 3. are its indicators (its mean and variance)

the others are all rv's

• 1. Is the Sample mean
• 4. Is the Sample deviance (a function of the Sample variance)
• 6. Is an estimator of $\theta$, a function of the data (in many cases 6. can be the same as 1.)