1. Two dice are tossed at random

a) Create a sample space matrix for the discrete random variable X= the sum of two tossed dice.

b) Create the probability mass function for X= x (the possible sums) and p(x) in a table. Does your pmf meet the fundamental requirements for a discrete pmf? Explain.

c) Find the expected value E(x) and the standard deviation Os by hand. Keep four decimal places. Show your work.

d) Enter the x values in L1 and the p(x) values in L2. Find the expected value E(x) and the standard deviation Ox using the STAT CALC 1: 1-Var Stats key sequence in your TI 84 to confirm your hand calculation. You must specify both lists where your data is entered, e.g., L1 and L2. Do the results match? If not, then keep trying.

e) Create the cumulative distribution function in table form.
2. Let the random variable X represent the score on an Engineering Statistics test. For X= x: 0, 28, 34, 58, 70, 85, 93, 100, the corresponding probabilities p(x) are 0.02, 0.01, 0.17,
0.24, 0.26, 0.15, 0.10, 0.05, respectively.

a) Does the given pmf meet the fundamental requirements for a discrete pmf? Why or
why not?

b) Enter the x values into L1 and the p(x) values into L2 to find the expected value and
standard deviation of the distribution. Explain your results.

c) Turn on Plot 1 to create a histogram on the TI 84. Select the histogram plot type. The XList is L1 and the Freq is L2. Make sure these are appropriately set. Unfortunately, ZOOM 9 does not work for a histogram. You must set up the WINDOW as follows: XMIN = 0, XMAX= 110, Xscl= 10, Ymin= 0, Ymax= 0.3, Yscl= 0.1. Xres= 1. Check these settings and your data and be sure these settings make sense to you. In the future, you should know how to set up the WINDOW on your own. Create the histogram. Use the TRACE and arrows to confirm the height of each bar. Place a picture of your histogram or draw it here and add the bar heights to each bar.

3. Suppose a component has a failure rate of 5% Let X denote the number among 12 randomly selected components. Use the 184 to find the following. Probability distribution functions are found under DISTR (2nd VARS). Show your work by showing the command(s) you enter. Round to the 10,000ths place. Verify that X is a binomial random variable, explaining using the four criteria for a binomial pmf.

a) Find the probability that 0 components fail.

b) Find the probability that exactly 10 components fail.

c) Find the probability that at most 2 components fail.

d) Find the probability that at least 2 components fail.

e) Find the probability that between 2 and 5 components fail P(2 < X s 5).

f) Find the mean, variance, and standard deviation for this binomial pmf.

Explain how your answers to a) e) make intuitive sense based on your answers to f).

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