Word problem based on Bernoulli trials

Question

I am studying for my final exam. Following is one of the question model that will be on the final:

Given the probability of a head is $p$ and tails $q$ and a random variable giving number of heads and an average. Calculate expectations and variances using definitions and theorems.

Can someone give me an actual word problem that will look like the above model and an explanation how to do it please? It will help me greatly. Thank you.

Answer 1

I won't give specific problems - as a tutor I find it more constructive for my students to understand the principles rather than memorizing patterns.

The binomial distribution is one of the most fundamental probability distributions out there. It is built from multiple Bernoulli experiments, each of which has only two outcomes - a success and a failure. You should remember the following about the binomial distribution:

• Each of the $n$ trials are independent - that is, one trial does not affect the outcome of another.
• The probability of success, $p$, is the same for all $n$ trials. As a consequence, the failure probabilities $q$ also remain the same.
• The probabilities come from the binomial coefficients $\binom{n}{k}$. Know how to compute these.
• For a binomial random variable $X$ with $n$ trials and success probability $p$, $E(X) = np$ and $V(X) = npq$.
• The words "success" and "failure" are relative and do not have to correspond to neccessarily good and bad events. Think about what you wish to compute - if you wish to determine the number of failed free throws given that a player will miss 53% of them, then $p = 0.53$.

Again, the key is to identity the proper parameters. Do some exercises which can easily be found on the internet or your text, and understand how the computations are done.

Answer 2

I'd add to Sean's answer, that in addition to rote memorisation of the expectation and variance, you should be able to derive them from definitions and theorems. (aka first principles.)

In this case, make use of indicator random variables, and the Linearity of Expectation and Bilinearity of Covariance.

You have a series of $n$ Bernoulli experiments, let that be $\{X_i\}_n$, where $X_i$ is the indicator for success. ie: $\mathsf P(X_i=1)=p ~,~ \mathsf P(X_i=0)=q$.

Hence the count of successes is $X=\sum_{i=1}^n X_i$.   Then since $\{X_i\}_n$ are iid random variables, that implies:

$$\begin{align}\mathsf E(X)~&=~ n\,\mathsf E(X_1) \\ &= n~(1\cdot\mathsf P(X_1=1)+0\cdot \mathsf P(X_1=0)) \\ &=~ np \\[2ex] \mathsf{Var}(X)~&=~ n\mathsf {Var}(X_1)\require{cancel}{+\cancelto{0}{n(n-1)\,\mathsf {Cov}(X_1,X_2)}}\\ &= n(\mathsf E(X_1^2)-\mathsf E(X_1)^2)\\ &=~ n (p-p^2)\\ &=~ npq\end{align}$$

Answer 3

It looks like the problems you should be able to solve look like the three below. Once you figure out the pattern, you can make up your own, just like I made up these. To solve them, you need to identify $p$ and $n$. It's easy if you follow the advice in the other answers. You should also verify that similar problems were assigned to you during the course. Otherwise, practice with these problems, useful as it may be, may not be sufficient preparation for your exam.

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