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Conditional expectation of continuous random variables: Practice

Problem 1. An unfair coin is tossed \(n\) times and all \(n\) tosses resulted in heads. What is the probability that the next toss will be head? Assume that the probability \(p\) of getting heads is a random variable with uniform \([0,1]\) distribution.

Problem 2. Let \(X\) be a random variable uniformly distributed on \([0, \pi]\). Find \(\mathbb E[X|\sin{X}]\).

Problem 3. Assume that \(X_1\), \(X_2\), \(X_3\), \(\dots\) are IID with uniform distribution on \([0,1]\). Let \(N\) be the smallest integer for which \[X_1+X_2+\cdots+X_N > 1.\] Evaluate \(\mathbb E\left[N\right]\).

Problem 4. Assume that \(X_1\), \(X_2\), \(X_3\), \(\dots\) are IID with uniform distribution on \([0,1]\). Let \(N\) be the smallest integer for which \[X_1+X_2+\cdots+X_N > 1.\] Evaluate \(\mathbb E\left[X_N\right]\).