Part 1:

Flipping a coin is a familiar exercise that can be used to demonstrate many of the properties of probabilities. In this assignment, we will be looking at flipping a coin three times and recording whether the coin lands heads up or tails up on each of the three flips. Note that since the same coin will be flipped 3 times, the order in which heads and tails shows is important.

1. It is noted in the problem above that a single coin will be flipped 3 times and that the order in which heads and tails show up is important. Based on this description of the problem, how many possible outcomes are there in the sample space?
2. Is this a uniform sample space? Justify your answer by specifically addressing the outcomes in the sample space and the definition of a “uniform” sample space.
3. Let X represent the number of heads in three separate tosses of a single fair coin (the situation described in the original problem and #1 above). Based on this definition of X, what are the possible values of X, and which outcomes from the sample space correspond to each of these values? You do not need to justify your answer.
4. Why is it possible for the sample space of a probability experiment to be a uniform sample space, yet the probabilities for the different values of X are not equal?

Part 2:

There are two ways to describe the define the distribution of a discrete random variable: using the probability mass function or the cumulative distribution function. For this discussion, let X represent a discrete random variable with probability mass function, f(x)$f\left(x\right)$, and cumulative distribution function F(x)$F\left(x\right)$.

Describe in your own words how the probability mass function for a discrete random variable is related to its cumulative distribution function. In your description, you should include what each function represents, a formula that relates the two functions (you can be creative with the notation if necessary), and an explanation in your own words of what the formula tells us about the relationship between the two functions.

And after that, I will send you a groupmate response for part 2 and evaluate it based on:

1. Did your groupmate completely answer the question asked in their initial post including all the specifics requested? Point out how each part of their answer corresponds to the specific details requested and identify each detail that is missing.
2. Did your groupmate answer the question correctly? If yes, then point out one particular part of their description that you found to be particularly helpful in demonstrating the relationship between the probability mass function and the cumulative distribution function. If no, then identify all parts of their description that are incorrect, not entirely correct, or misleading and suggest how they could have better addressed each incorrect part of their description.

The evaluation is meant to be done after I have been assigned a groupmate which will be after 12am Eastern time tonight.