Find the cumulative distribution function (cdf) for an ...



Beta distribution problems on last sample test.

Math 309 Test 4 Carter Name_____________________________

Show all work in order to receive credit. 11/29/01

1. Consider a die with three equally likely outcomes. A pair of such dice is rolled. Let X denote the sum on the pair of dice and Y denote the “larger” number. The joint probability distribution is in the chart.

a) P(X ( 4, Y ( 2)

b) P( X = 4)

c) P(X ( 4)

d) P(Y = 2 | X = 4)

2. Consider the joint density function: [pic]

a) Find P(X < ¾, Y < ¼).

b) Find P(X < ¾, Y < ½).

c) Find P(X < ¾ | Y < ½).

d) Find the conditional density function for X given Y = 1/2.

e) Find P(X < 3/4 | Y = 1/2 ).

3. Consider the joint density function: [pic]

a) Find the marginal density functions f1(x) and f2(y).

b) Are X and Y independent? Justify.

c) Find P( X < Y) . (Set-up is sufficient.)

d) Set-up an integral that gives E[X-Y].

4. Two friends are to meet at a library. Each arrives randomly at an independently selected time within a fixed one-hour period and agrees to wait no longer than 15 minutes for the other. Find the probability that they will meet. (Set up is sufficient.)

5. Explain the statement, “While covariance measures the direction of the association between two random variables, the correlation coefficient measures the strength of the association." Examples are appropriate in your comments.

6. Select one of the following.

a) For either the discrete or continuous case, show that if X and Y are independent, E[XY] = E[X]E[Y].

b) Show that if X and Y are independent, Cov(X, Y) = 0. (Assume (a) & use the definition, cov(X,Y) = E[(X-μx) (Y-μy)].)

c) Let Y1, Y2, …, Yn be independent random variables with E(YI) = μ and V(YI) = σ2. Show E[X] = nμ and V(X) = nσ2 where X = Y1 + Y2 + … + Yn.

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Answers/Solutions: Believed accurate, but not guaranteed! For some I have given answers. Others have an intermediate step in the solution as I felt it would be more beneficial than just the answer‮ㄍ‮潙⁵敮摥琠湫睯愠潢瑵琠敨搠敩‮獁畳敭琠慨⁴桴⁥楤⁥慨⁳睴鈱ⱳ琠潷㈠玒愠摮琠潷㌠玒മउ൘ፙ䔠䉍䑅䔠畱瑡潩.

1. You need to know about the die. Assume that the die has two 1’s, two 2’s and two 3’s.

X

Y[pic]

5/9, 3/9, 6/9, 1/3

2a.[pic] b. [pic]

c. [pic]

d. [pic]

e. [pic]

3a.[pic]

b. X and Y are independent since [pic].

c. [pic]

d. [pic]

4. [pic][pic]

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