HW5_solutions_b

HW5_solutions_b - 23 45 13 26 HW#5, Solutions 2011) AMS 311...

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6 3 5 3 2 1 4 2 HW#5, Solutions AMS 311 Probability Theory (Spring 2011) Part 1 (10 points): PRINT your name and StonyBrook ID ( Last, First, #ID ) on the top on your work sheet. Staple your HW if more than 1 pages. Part 2: Problems (90 points) Write down your work step by step to get full score. (1). (10 points) Let X be the size of claims 10 years ago, Y the size of claims today, so we have Y=2X. (Proper notations for random variables are important!) X~exp(λ), we can determine λ first. We know P(X<1000)=F X (1000)=1-exp(-1000λ)=0.25, so λ=-ln(0.75)/1000=2.88E-4 Hence, P(Y<1000)=P(2X<1000)=P(X<500)=F X (500)= 1-exp(-500λ)=0.134 (2). (15 points) (a) Let X be the distance between any randomly selected tree and the tree nearest to it. X~exp(λ). We know EX=40=1/ λ, so λ=1/40. P(X>30)=1-F(30)= exp(-30λ)=0.472 (b) P(X>80|X>50)=P(X>30)=0.472 (because X is memoryless) (c) to find minimum t, so that P(X>t)≥0.5 P(X>t)=1-F(t)= exp(-t/40)≥0.5, so -t/40≥ln(0.5), t≤-40*ln(0.5)=27.7 Hence, the minimum distance is 27.7 feet. (3). (25 points) (a) (b) Homework is due at the beginning of class on its due date. You must be in class and on time to submit your homework. No homework will be accepted via email. No late homework will be accepted. Your lowest homework grade will be dropped before computing your average.
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This note was uploaded on 05/09/2011 for the course AMS 311 taught by Professor Tucker,a during the Spring '08 term at SUNY Stony Brook.

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HW5_solutions_b - 23 45 13 26 HW#5, Solutions 2011) AMS 311...

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