1. Consider two random variables X and Y with joint PMF given in the table Find P(X = 2, Y <2) Find marginal PMF of X and Y Find P(Y=2 |X= 1) Are X and Y independent a. b. 3 С. 1 1/6 0.25 0.125 d. 2 0.125 1/6 1/6

Test the claim about the difference between two population means mu 1μ1 and mu 2μ2 at the level of significance alphaα. Assume the samples are random and​ independent, and the populations are normally distributed.   ​Claim: mu 1μ1equals=mu 2μ2​; alphaαequals=0.01 Population​ statistics: sigma 1σ1equals=3.3​, sigma 2σ2equals=1.5 Sample​ statistics: x overbar 1x1equals=17​, n1equals=29, x overbar 2x2equals=19, n2equals=28 Determine the alternative hypothesis.   Upper H Subscript aHa​: mu 1μ1 ▼   not equals≠ greater than or equals≥ less than or equals≤ greater than> less than< mu 2μ2 Determine the standardized test statistic.   zequals=nothing ​(Round to two decimal places as​ needed.) Determine the​ P-value.   ​P-valueequals=nothing ​(Round to three decimal places as​ needed.) What is the proper​ decision?     A. Reject Upper H 0H0. There is enough evidence at the 1​% level of significance to reject the claim.   B. Fail to reject Upper H 0H0. There is not…

Suppose you have a normally distributed population such that o? = 7. Using the sampling distribution of S?, we find that from a random sample of size n = 11, s² = 16.053. Does our population variance seem || reasonable? Note: We will consider our population variance reasonable if our x value falls within the interval x0.975 (df), x'o.025 (df)]. 1) Find the values of the interval [xo.975(df), x²o.025(df)]. x°0.975(df) = (Round to 3 decimals.) x'0.025(df) - (Round to 3 decimals.) 2) Find the value of x that comes from the data you collected in the problem statement. (Round to 3 decimals.) 3) Is the population variance reasonable? Yes O No

Match the following.

5. Let X₁ and X₂ be iid Poisson() random variables with pmf p(x) = H'e exp{p(e-1)), for tER. Let Y=X₁+X₂. a. b. What is the distribution of Y? What are E[Y] and Var[Y]? for x 0,1,2,..., and mgf 9 X!

7. If the r.v.'s X and Y have bivariate normal distribution with parameters ₁ = 3.2, µ2 = 1.44, 0₁ = 12, 02 = 16 and p = 0.7, determine the following quantities: (a) E(X), E(Y), Var(X), Var(Y), p(X, Y), and Cov(X, Y). (b) the distribution of X and the distribution of Y. (c) the probabilities P(0.8 14). (d) the conditional distribution of Y, given X = 3.8. (e) the conditional distribution of X, given X = 10. (f) the probabilities P(X > 3.2|Y = 10), P(Y < 12|X = 3.8). (g) E(X|Y = 10), E(Y|X = 3.8), Var(X|Y = 10), Var(Y|X = 3.8).