Please show all steps & explaination if needed Consider rolling a standard 6 sid

Question

Please show all steps & explaination if needed

Consider rolling a standard 6 sided red dice and a 6 sided blue dice at the same time. (a) List all possible outcomes for this experiment (i.e sample space). (b) Run the experiment three times and write down your results. (c) Assume we are interested in the discrete random variable which represents the lowest of the two numbers rolled. Derive the probability mass function for this situation. (d) Determine the expected variable and standard deviation for this discrete random variable (in a table - i.e by hand) (e) Determine the percentage of data within 1, 2 and 3 standard deviations of the mean and comment on the results.

Explanation / Answer

a. There are 36 possible outcomes - 6 possibilities for red dice and 6 for blue dice.

b. We run the experiment 3 times and get 3 outcomes as (1,3), (2,5), and (5,3)

c. Let X denote the lowest of the two numbers.

From the above table we get:

P(x=1) = 11/36

P(x=2) = 9/36

P(x=3) = 7/36

P(x=4) = 5/36

P(x=5) = 3/36

P(x=6) = 1/36

d. Expected value = 1.11/36+2.9/36+3.7/36 + ..6.1/36 = 91/36

Variance = E(X^2) - E(X)^2 = 1.11/36 + 4.9/36 + 9.7/36 + ..+36.1/36 - (91/36)^2 = 301/36 - (91/36)^2 = 2555/36^2 = 1.97

Standard deviation = sqrt(1.97) = 1.404

Red Blue Lower 1 1 1 1 2 1 1 3 1 1 4 1 1 5 1 1 6 1 2 1 1 2 2 2 2 3 2 2 4 2 2 5 2 2 6 2 3 1 1 3 2 2 3 3 3 3 4 3 3 5 3 3 6 3 4 1 1 4 2 2 4 3 3 4 4 4 4 5 4 4 6 4 5 1 1 5 2 2 5 3 3 5 4 4 5 5 5 5 6 6 6 1 1 6 2 2 6 3 3 6 4 4 6 5 5 6 6 6

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