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Consider an infinite sequence of independent (possibly unfair) coin flips where the probability of each being heads is pe[0,1]. We define the family of random variables X to be the number of heads after k flips. i) Using the fact that X, is binomially distributed with X, ~ B(n, p), calculate P(X5 = 2). ii) Let k>0 be even, and p =5. Calculate P(X >5). Hint: you might want to also consider P(X <5). We now define the random variable Y as the length of the run of either all heads or all tails at the start of an infinite series of coin flips. For example Y = 4 if the sequence of coin flips start with HHHHT... or TTTTH.... iii) Give an expression of P(Y = k) in terms of p and k. iv) Using your answer to the previous question, show that 1- 2p + 2p² p(1 - p) E(Y) = You will probably want to use the fact that: 1 ia'-1 (a – 1)2 i=1

Consider an infinite sequence of independent (possibly unfair) coin flips where the probability of each being heads is pe[0,1]. We define the family of random variables X to be the number of heads after k flips. i) Using the fact that X, is binomially distributed with X, ~ B(n, p), calculate P(X5 = 2). ii) Let k>0 be even, and p =5. Calculate P(X >5). Hint: you might want to also consider P(X <5). We now define the random variable Y as the length of the run of either all heads or all tails at the start of an infinite series of coin flips. For example Y = 4 if the sequence of coin flips start with HHHHT... or TTTTH.... iii) Give an expression of P(Y = k) in terms of p and k. iv) Using your answer to the previous question, show that 1- 2p + 2p² p(1 - p) E(Y) = You will probably want to use the fact that: 1 ia'-1 (a – 1)2 i=1

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 32E
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Consider an infinite sequence of independent (possibly unfair) coin flips where the probability of each being heads is pe [0,1]. We define the family of random variables X to be the number of heads after k flips. i) Using the fact that X, is binomially distributed with X„ ~ B(n,p), calculate P(X5 = 2). ii) Let k> 0 be even, and p = 5. Calculate P(X > 5). Hint: you might want to also consider P(Xr <5). We now define the random variable Y as the length of the run of either all heads or all tails at the start of an infinite series of coin flips. For example Y = 4 if the sequence of coin flips start with HHHHT... or TTTTH.... iii) Give an expression of P(Y = k) in terms of p and k. iv) Using your answer to the previous question, show that 1– 2p + 2p² p(1 -р) E(Y) = %3D You will probably want to use the fact that: 1 ia (а – 1)2 i=1
Transcribed Image Text:Consider an infinite sequence of independent (possibly unfair) coin flips where the probability of each being heads is pe [0,1]. We define the family of random variables X to be the number of heads after k flips. i) Using the fact that X, is binomially distributed with X„ ~ B(n,p), calculate P(X5 = 2). ii) Let k> 0 be even, and p = 5. Calculate P(X > 5). Hint: you might want to also consider P(Xr <5). We now define the random variable Y as the length of the run of either all heads or all tails at the start of an infinite series of coin flips. For example Y = 4 if the sequence of coin flips start with HHHHT... or TTTTH.... iii) Give an expression of P(Y = k) in terms of p and k. iv) Using your answer to the previous question, show that 1– 2p + 2p² p(1 -р) E(Y) = %3D You will probably want to use the fact that: 1 ia (а – 1)2 i=1
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