By Sheldon M. Ross: An Elementary Introduction to by Sheldon M. Ross

By Sheldon M. Ross

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Thus, we are supposing that the process values change only at times that are integral multiples of , and that at each change point√the value of the process either increases or decreases by the amount σ√ , with ). the change being an increase with probability p = 21 (1 + μσ As we take smaller and smaller, so that changes occur more and more frequently (though by amounts that become smaller and smaller), the process becomes a Brownian motion with drift parameter μ and variance parameter σ 2 . Consequently, Brownian motion can be approximated by a relatively simple process that either increases or decreases by a fixed amount at regularly specified times.

Random Variables and Expected Values 13 In other words, the variance measures the average square of the difference between X and its expected value. 3f Find Var(X ) when X is a Bernoulli random variable with parameter p. Solution. 3c), we see that (X − E[X ])2 = Hence, (1 − p)2 with probability p with probability 1 − p. p2 Var(X ) = E[(X − E[X ])2 ] = (1 − p)2 p + p 2 (1 − p) = p − p 2. 7)) = E[a 2 (X − E[X ])2 ] = a 2 Var(X ). 8) Although it is not generally true that the variance of the sum of random variables is equal to the sum of their variances, this is the case when the random variables are independent.

10 Two players play a tennis match, which ends when one of the players has won two sets. Suppose that each set is equally likely to be won by either player, and that the results from different sets are independent. Find (a) the expected value and (b) the variance of the number of sets played. 11 Verify that Var(X ) = E[X 2 ] − (E[X ])2 . Hint: Starting with the definition Var(X ) = E[(X − E[X ])2 ], square the expression on the right side; then use the fact that the expected value of a sum of random variables is equal to the sum of their expectations.

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