# Probability

We have k coins. The probability of Heads is the same for each coin and is the realized value q of a random variable Q that is uniformly distributed on [0,1]. We assume that conditioned on Q=q, all coin tosses are independent. Let Ti be the number of tosses of the ith coin until that coin results in Heads for the first time, for i=1,2,…,k. (Ti includes the toss that results in the first Heads.)

You may find the following integral useful: For any non-negative integers k and m,

∫10qk(1−q)mdq=k!m!(k+m+1)!.
Find the PMF of T1. (Express your answer in terms of t using standard notation.)

Find the least mean squares (LMS) estimate of Q based on the observed value, t, of T1. (Express your answer in terms of t using standard notation.)

We flip each of the k coins until they result in Heads for the first time. Compute the maximum a posteriori (MAP) estimate q^ of Q given the number of tosses needed, T1=t1,…,Tk=tk, for each coin. Choose the correct expression for q^.

q^=k−1∑ki=1tiq^=k∑ki=1tiq^=k+1∑ki=1tinone of the above

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2. These questions are from online courses, and you need to study and know the answers to get a credit.

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3. 3. second choice

q = k/sum(k, i=1) t_i

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4. 1. part I: use conditional probability to solve the PMF

2. part II: use the mathematical definition of expectation (integration) to solve for the estimator

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5. Can anonymous be more specific (please!)

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6. can anyone provide the answers??????

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7. 1) 1/(t*(t+1))
2) 2/(t+2)

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