# STA – STA TEST

### STA TEST

STA – 本题是一个利用STA进行练习的代做, 对STA的流程进行训练解析, 是比较典型的STA等代写方向 ``````Name
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``````Student Number
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### STA 256 f2019 LEC0101 Test 3

``````Tutorial Section (Circle One)
``````
``````TUT0101 TUT0102 TUT0103 TUT0104 TUT
Mon. 3-4 Mon. 4-5 Mon. 5-6 Mon. 6-7 Wed. 4-
DH 2080 DH 2080 IB 360 IB 240 IB
Ali Dashvin Dashvin Ali Marie
TUT0106 TUT0107 TUT0108 TUT0109 TUT
Wed. 5-6 Fri. 9-10 Fri. 10-11 Fri. 10-11 Fri. 11-
IB 360 IB 200 DH 2070 DV 3093 DH 2070
Marie Crendall Ali Cendall Ali
TUT0111 TUT0112 TUT0113 TUT0114 TUT
Fri. 11-12 Fri. 12-1 Fri. 4-5 Fri. 5-6 Fri. 6-
DV 3093 DV 2070 DV 3093 IB 360 IB 360
Crendall Crendall Karan Karan Karan
TUT0116 TUT
Wed. 11-12 Wed. 12-
DH 2070 IB 260
Ana Ana
``````
``````Question Value Score
1 20
2 15
3 25
4 20
5 20
``````
``````Total = 100 Points
``````
1. Let the random variablesXandY have joint density
``````fX,Y(x,y) =
``````

#### {

``````24 xy For 0x 1 , 0 y1 andx+y 1
0 Otherwise
``````
``````(a) (3 points) Sketch the region of thex,yplane where the joint density is non-zero.
``````
``````(b) (8 points) Find the marginal densityfX(x). Show your work. Be sure to specify
where the density is non-zero.
``````
``````(c) (1 point) You know the marginal densityfY(y) by symmetry. Just write it down.
Be sure to specify where the density is non-zero.
``````

``````(e) (6 points) Give the conditional densityfY|X(y|x). Be sure to specify where the
density is non-zero.
``````
1. (15 points) Let X and Y be independent, discrete random variables. Show that E{g(X)h(Y)}= E{g(X)}E{h(Y)}. BecauseX andY are discrete, you will add rather than integrating to do this question. Be very clear about where you use inde- pendence. Draw an arrow pointing to where you use independence, and write This is where I use independence.
1. LetX 1 andX 2 be independent normal random variables with= 0 and^2 = 1. Let Y 1 =X 1 X 2 andY 2 =X 1 +X 2.
``````(a) (10 points) Calculate the joint density ofY 1 andY 2. Show your work, andcircle
your final answer. The next part of this question will go better if you simplify
``````

Continue Question 3 if necessary.

(b) (10 points) Find the marginal density ofY 1 =X 1 X 2.

``````(c) (5 points) The distribution ofY 1 is one of the common distributions on the
formula sheet. Identify it by name and give the value(s) of the parameter(s).
``````
1. LetXhave a binomial distribution with parametersnand.
``````(a) (12 points) Derive the moment-generating function ofX. Show your work.Cir-
you force your answer to come out right by making a convenient mistake, you
will get a zero for this part.
``````
``````(b) (8 points) Use the moment-generating function to findE(X). Show your work.
``````X 1 Gamma (= 1,= 1) X 2 Gamma (= 2,= 1) X 3 Gamma (= 3,= 1)
``````Find the distribution ofY =X 1 +X 2 +X 3. Show your work. It is one of the common
``````Total Marks = 100 points