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Tik-61.183 Special course on information technology III

Randomized algorithms
Spring 2000
Heikki Mannila

Exercises 1, due February 9, 2000

1.
The variance D2(X) of a random variable X is the expectation of the random variable (X-E(X))2, i.e., D2(X) = E((X-E(X))2). The variance is also denoted by $\sigma_X$, or by var[X]. Show that D2(X) = E(X2) - (E(X))2.
2.
Let X(n) be the number of tails obtained when flipping a fair coin n times. What is E(X(n))? What is D2(X(n))?
3.
Let X(n,p) be the number of heads obtained when flipping a biased coin (probability of head = p) n times. Compute E(X(n,p)) and D2(X(n,p)).
4.
Find Markov's inequality from a textbook and give an informal justification of it. State Chebyshev's inequality.
5.
Approximate (either analytically or by simulation) the probability of the following event: ``When flipping a fair coin 1000 times, the number of heads is less than 450''.


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