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Week_2_Practice_Problems_Solution

Week_2_Practice_Problems_Solution - Week 2 Practice...

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Week 2 Practice Problems (Review discrete random variables and their properties) 1. Suppose Y is a discrete random variable with probability function y 0 1 2 3 ( ) p y 1/10 2/10 3/10 4/10 a) Find and [ ] E Y [ ] sd Y E[Y] = 0×1/10+1×2/10+2×3/10+3×4/10 = 2 V[Y] = 0×1/10+1×2/10+4×3/10+9×4/10 –E[Y]^2 = 5-4=1 so sd [Y] = 1 b) Suppose are n independent copies of Y as in a) and 1 2 , ,..., n Y Y Y 1 2 ... n Y Y Y Y n + + + = is the average. Find [ ] E Y and [ ] sd Y 1 1 [ ... ] [ ] ... [ ] [ ] 2 n n E Y Y E Y E Y E Y n n + + + + = = = 2 1 1 2 2 1 1 [ ] ( [ ... ]) ( [ ] ... [ ]) n n sd Y Var Y Y Var Y Var Y n n = + + = + + 1 n = So [ ] 1/ sd Y n = 2. In an experiment to test the effect of a pre-treatment of seeds on the germination rate, each seed can be considered a Bernoulli trial a) What are the conditions that must be satisfied for Bernoulli trials in this case. A seed germinates or not each seed has the same probability to germinate all these tries are independent b) Suppose 20 seeds are treated and the germination rate is 1 π . What is the probability that 16 seeds germinate?

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