Reference no: EM132855750

You are the engineer of record (i.e. responsible) for a stream restoration pilot project. There are four (4) streams of interest. Last Fall you built experimental anti-erosion systems on each of the streams. You hope that the streams will flood this Spring, and over top their banks, so that you can see how the engineered systems respond. From experience, and the historical record, your boss estimates that the odds of each stream overtopping its banks this Spring (i.e. flooding) is 3 to 1 (i.e. 75 percent chance) for each of the streams. Since the streams are geographically separate - spread out around the state with non-overlapping watersheds - we will model the probability of any one stream flooding as statistically independent from that for any of the other streams. If a stream flooding is called a success (designated by an "S"), and everything else is a failure (designated by an "F"), one possible outcome of this "random experiment" is the sequence (SSFS). Let the discrete random variable X be the number of successes; thus, for the outcome (SSFS), X = 3. The discrete random variable X can take on five possible values: {0, 1, 2, 3, 4}.

a-Exhaustively enumerate all possible outcomes (i.e. every possible sequence of "S" and "F"). In this case, order matters. There are two possibilities for each stream (S or F), and there are four streams, so there are 24 = 16 possible outcomes.

b-What is the probability of each outcome? Remember that Pr(S) does not equal Pr(F).

c-Enumerate the associated values of X. Remember, X is the number of successes.

I found that building a small table in Excel to be an effective way to attack this problem. The table has 16 rows, one row for each event, and 3 columns. The first column identifies the event (e.g. SSFS) - this is part (a). The second column contains the X associated with the event - this is part (c). The third column contains the probability of the event - this is part (b). The remaining parts can then be answered using the values in this table.

d-Give the probability mass function for X. You may do this part with a table or with a simple graph.

e-Give the cumulative distribution function of X. You may do this part with a table or with a simple graph.

f-What is the expected value of X?

g-What is the variance of X?

h-What is the standard deviation of X?

i-What is the mode of X?

j-What is the median of X?

For the purposes of this specific problem, carry all of the digits through each of the questions (with the exception of the standard deviation).