Let X be the number of packages being mailed by a randomly selected customer at a certain shipping facility. Suppose the distribution of X is as follows. X 1 2 3 4 p(x) 0.2 0.4 0.3 0.1 (a) Consider a random sample of size n = 2 (two customers), and let X be the sample mean number of packages shipped. Obtain the probability distribution of X. 1 1.5 2 3 4 X P(X) 0.04 0.16 0.28 X .46 2.5 .28✔✔✔ .17 (b) Refer to part (a) and calculate P(X ≤ 2.5). .76 (c) Again consider a random sample of size n = 2, but now focus on the statistic R = the sample range (difference between the largest and smallest values in the sample). Obtain the distribution of R. [Hint: Calculate the value of R for each outcome and use the probabilities from part (a).] R 0 1 2 3 P(R) 28 2 3.5 ✓ .06 ✔ .01 .04 (d) If a random sample of size n = 4 is selected, what is P(X ≤ 1.5)? [Hint: You should not have to list all possible outcomes, only those for which X < 1.5.]

Let X be a random variable that has a Poisson distribution. If the probability that X is greater than 0 is 0.9967. Then the mean, variance, and the standard deviation of the probability distribution of X are, respectively, (Choose one from the following which is closest to the correct answer) O c. 4.3, 2.5, 1.6. O a. 5.7, 5.7, 2.4. O d. 5.7, 2.5, 6.0. O b. 5.5, 2.5, 1.6. O e. 10.0, 4.3, 1.6.

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The probability distribution of the number of students absent on Mondays, is as follows: X O 1 2 3 4567 f(x) 0.02 0.03 0.26 0.34 0.22 0.08 0.04 0.01 (a) What is the probability that more than 3 students are absent. (b) Compute the expected value of the random variable X. Interpret this expected value. (c) Compute the variance and standard deviation of the random variable X. (d) Compute the expected value and variance of Y : (e) Compute the covariance Cov(X, Y) = 7X+ 3. (f) What is the value of the ratio Cov(X,Y)/Var(X) ? (g) What is the value of Cov(X, Y)/Var(X) for Y = ß0 + ß₁X ? (For this exrcise, you can use excel. No need to show all calculations, just specify the formulas and present the final values.) (a) (b) (c) (d) (e) (f) (g)