Question

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.)

For t=1,…, pT1(t)=- unanswered
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.)

E[Q∣T1=t]=- unanswered
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.

Answer 1

Answer:

Step-by-step explanation:

The question is not in correct order ; so the first thing we are required to do is to put them together in the right form to make it easier to proof; having said that. let's get started!.

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

The next  equation goes thus : $\int\limits^1_0 \ \ q^k (1-q)^m dq = \frac{k!m!}{(k+m+1)!}$

a. Find the PMF of $T_1$ .   (Express your answer in terms of t using standard notation.)

For t=1,2…,

$p_T_1$(t)= $\frac{1}{(t*(t*1))}$  by using conditional probability to solve PMF

b)  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.)

By using the mathematical definition of integration to solve the estimate of Q:

$E|Q|T_1=t|= \frac{2}{t+2}$

c)  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.

The MAP estimate of  q is :

$\bar q = \frac{k}{\sum^k_{i=1}}t_i$