final_2005_solution

final_2005_solution - SECTION A: SHORT ANSWER SECTION:...

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SECTION A : SHORT ANSWER SECTION :( each question is worth 3 marks , total marks 36 ) [3] a. Suppose X has a normal distribution with mean 2 and variance 4. Find the constant c so that P | X 2| c 0.99 P c X 2 c P c 2 X 2 2 c 2 so c 2 2.57 or 2.58. Solve for c [3] b. A random variable X has probability function x 012 P X x 0.5 0.3 0.2 . Find the moment generating function of X . M t E e Xt 0.5 0.3 e t 0.2 e 2 t [3] c. Use an expression for P A B (the additivity rule of probability) to prove that P A B P A P B 1 (Hint: this can be done in a couple of lines.) P A B P A P B P A B Therefore P A B P A P B P A B P A P B 1 [3] d. If P A 0.6, P B 0.5 and P B | A 0.5, find P A B .Are A , B independent? P AB P A P A B P A P A P B | A 0.6 P A B P A P B P A B 0.8 Since P AB P A P B the events are independent.
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[3] e. Suppose X and Y are discrete random variables with joint probability function f x , y 1 21 x 2 1 y for x 1,2 and y 1,2,3 and otherwise f x , y 0. Are X and Y independent? (explain) Try any pair of values. For example f 1,1 1 21 but f 1 1 6 21 , f 2 1 5 21 Since f f 1 1 f 2 1 they are NOT INDEPENDENT0.4 in [3] f. Suppose X , Y are independent normal random variables with E X E Y 1and Var X Var Y 5. Find P X 2 Y 3 1 . E X 2 Y 3 1 2 3 2 Var X 2 Y 3 25 P X 2 Y 3 1 P X 2 Y 3 2 5 1 2 5 P Z 0.2 where Z is standard normal .5793 [3] g. Suppose X has an exponential distribution with probability density function f x 2 e 2 x ,fo r x 0 Find P X 5| X 3 and compare with P X 2 .
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final_2005_solution - SECTION A: SHORT ANSWER SECTION:...

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