Question

A random experiment consists of throwing a coin marked with the labels "Face" and "Cross". Suppose the probability of obtaining each of these results is p and 1-p, respectively. Calculate and graph accurately the probability function and the function of distribution of variable X defined as follows:

X("Face")=3

X("Cross")=5

Answer 1

This is a simple problem related to

1. Development of the probability distribution function (PDF) for the given process

2. Trace the graph for the same.

Now let us first of all try to understand as ot what is gonig on in the process.

There is a coin (biased or fair ) with two sides -say Face and Cross.

The probability of getting a face in a single toss is p

The probability of not getting a face will hence be 1-p

Now,

One way to think about this is by thinking of the probability (p) of the coin landing on face to be a random variable.

Clearly the distribution for p is a uniform random variable over [0,1] (any value between 0 to 1)

After flipping the coin some number of times and collecting data, we can formulate our distribution accordingly.

Now let

If you don't know whether it is a fair coin to start with, then it isn't a dumb question at all. (EDIT) You ask if the coin will be biased towards Tails to account for the all of the heads. If the coin was fair, then the answer from tilper addresses this well, with the overall answer being "No". Without that assumption of fairness, the overall answer becomes "No, and in fact we should believe the coin isbiased towards heads.".

One way to think about this is by thinking of the probability pp of the coin landing heads to be a random variable. We can assume that we know absolutely nothing about the coin to start with, and take the distribution for pp to be a uniform random variable over [0,1][0,1]. Then, after flipping the coin some number of times and collecting data, we change our distribution accordingly.

Now let

where (α−1 )represents the number of face we've recorded and (β−1 ) the number of cross we have recorded .

The probability distribution function will be given by

where k is a constant.

Incidently this kind of a distribution is known by a special nomenclature called as Beta Distribution and is generally represented as

The denominator is simply a constant used to normalize the distribution and is given by  

Now we have got the function of the distribution and we now need to trace it.

To observe the holistic view of the curve for a fixed probability p and for varying we took few generic values of both and simulated the curve designs.

We get the given curves for various values of and for change in value of p from 0 to 1.

Now we are given with

so we inserted the pdf curve for the same too (shown in amber)

The cumulative distribution function will be given by another beta distribution as

The corresponding plot for the same will be