# How to find the estimator using random variables in statistics

I'm doing an assignment for homework in my statistics class. I'm having trouble really understanding what is going on when it comes to estimators, and what the estimator of something is given a random sampling. Here is the question:

Suppose a researcher collected $n = 86$ randomly selected rats and adds injects no cancer-causing substance (i.e. 0 ng/kg/day).

a. Let ($Y_1,Y_2,...,Y_{86}$) denote a random sample, where $Y_i \sim Bernoulli(p_2)$. What is a parameter of interest?

b. Interpret the parameter of interest in the context of this experiment.

c. What is an estimator of $p_2$ using the random sample?

d. Among the 86 rats collected, 9 rats developed tumors. What is the estimate of $p_2$ based on the proposed estimator?

e. If $p_2 = 0.1$, what is the probability distribution of $\sum_{i=1}^{86}Y_i$?

f. If $p_2 = 0.1$ what is the probability distribution of the estimator of $p_2$? Provide the PMF define on the support of the estimator.

My answers:

a. For part a, I'm assuming the parameter of interest is $p_2 = E(Y_i)$, which is the average of each Bernoulli random variable.

b. Since $p_2 = P(Y_i = 1)$, I think it should be interpreted as the probability of tumor development when a randomly selected rat receives 0 ng/kg/day of the cancer causing substance.

c. This is where I'm, having trouble. Since an estimator estimates $p_2$, shouldn't it just be the same as part a? So the estimator is $p_2 = E(Y_i) = \frac{1}{n}\sum_{i=1}^{n}Y_i$?

d. Wouldn't this just be equal to 9/86 rats? Even though I'm not using the estimator.

As for e and f, I'm unsure as to what I can use to get these answers. If anyone can help me with c-f or even post a helpful link for this kind of stuff I would appreciate it. I'm just very confused as to what I should be doing.

## 1 Answer

Your answer to (d) looks sensible but, as you say, it it is not consistent with your answer to (c). It is your answer to (c) which is wrong (though you have edited it while I am typing) as $E(Y_i)$ is not observable but is in fact what you are trying to estimate with the estimator.

An estimator is a rule for calculating an estimate of a given quantity based on observed data, and the rule you appear to have applied in (d) seems to be to calculate the observed proportion of rats which have developed tumours. So that should be your answer to (c): $\frac{1}{n}\sum_{i=1}^{n}Y_i$ rather than $E(Y_i)$

The answers to (e) and (f) are related to the sum of Bernoulli random variables.

• After reading the wikipedia link, I'm still kind of confused on how to solve e and f. For e, would it be a binomial distribution of $B(1,0.1)$? And for f, what is the "PMF define on the support of the estimator"? Thanks for your help by the way! – Alex May 12 '15 at 5:49
• Since there are $86$ rats, the number with tumours has a $B(86,0.1)$ distribution, with support on the integers from $0$ to $86$. You have to divide this by $86$ to estimate $p_2$ so the support of the estimator is of the form $\frac{n}{86}$ from $0$ to $1$ and on this support you have $\Pr(\hat p_2 = p) = {86 \choose 86p}0.1^{86p}0.9^{86-86p}$. – Henry May 12 '15 at 6:02
• So for e though, I'm just confused on how I can apply the binomial distribution formula which is: $\sum_{y=0}^{n}P(Y=y) = \sum_{y=0}^{n} \frac{n!}{y!(n-y)!}p^y(1-p)^{n-y} = 1$ according to my textbook. In this case I know n = 86 and p = 0.1, but how do I solve for y? Do I assume y = 1? – Alex May 12 '15 at 6:10
• Also what is the $p$ in $Pr(p_2 = p)$? – Alex May 12 '15 at 6:22
• If $y$ is some integer between $0$ and in this case $n=86$ then the distribution of $Y$ has probability mass function $P(Y=y) = \frac{n!}{y!(n-y)!}p^y(1-p)^{n-y}$. You are not trying to solve for $y$ but simply stating the probability of observing it. – Henry May 12 '15 at 6:27