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homework4

homework4 - Homework 4 36-325/725 due Friday Sept 21(1a Let...

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Unformatted text preview: Homework 4 36-325/725 due Friday Sept 21 (1a) Let be such that . Draw a graph of the cdf . (1b) Let . be a discrete random variable. Prove that (3) Let be a random variable such that . Suppose that has continuous, differentiable cdf , pdf and suppose that for all . Define and let . The function is called the hazard function. Think of as a lifetime; for example, the lifetime of a transistor or a light bulb or a person. (3a) Show that Use this result to give an interpretation of . (3b) Show that, for all , (3c) Suppose that . Show that (4) Let and let } (4a) Find the pdf for . Plot it in R. 1 G [email protected] % [email protected] g% [email protected] Qq [email protected] g p6&@ & [email protected] c fe6d w [email protected] y xw #iu kUv t o s 2 eq q YD Pm5#jHBhd gfd r p o n G l kUi @ c e @ c eX @ 2 Bd7 8 [email protected] @ c @ ) t ~ } `B% @ % [email protected] 'q G X { t \$Y |z % (2) Let have cdf . Let . Find the cdf of . GD @ [email protected] % CBA9 . Interpret this result. % % 20 4!31) (' & and . Use to find D P a X W UT R D b`YVEVSQP I [email protected] % 4!2308765 ) ) ! \$#" and . y <- e^x Draw a histogram of and compare it to the pdf from (3a). (5) Let (6) Let and are independent. Show that the distribution of where . Hint 1: You may use the following fact: If , and and are independent, then Hint 2: Note that (7) Let Find . The following are practice problems. Do not hand them in. [email protected] @ G c yi7 & y8 2 (8) Do the following questions two ways: using back of DeGroot. (8a) Let . Find . (8b) Let . Find . (8c) Let . Find such that [email protected] G mbS } Szs X @ W a x v } `vv ` y it ( t } y`wyy t v } } x ``y t v v g 2i i `3v ) e ) T ) @ e eX |83C|6) u [email protected] [email protected] Let be uniformly distributed on the unit disc . Find the cdf and pdf of . Hint: . Now use geometric reasoning. and assume that given that and using the tables at the ) x % a ) e @ eX |[email protected] W and and . . . (4b) Generate a vector Then: ) } e ) } } 1 } u X } ("W [email protected] } y`wyy v v X t R v y T y3h v v y`w t } } & d @ consisting of 10,000 random standard Normals in . X { t y\$Y i" y\$Y iz X { t X { t y\$Y i" e . is ys76) ) 42se @ c @ 0 & ) dd6 . . y`wyyiu v v t X { t y\$Y i" X { t y\$Y iz (8d) Let (8e) Let (8f) Let . Find . Find such that . Find 3 . ...
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