Monte Carlo Approximations – Introduction Suppose X1, X2,… is a sequence of independent and identically distributed random variables with unknown mean µ. Let The laws of large numbers indicate that for large n, Mn ≈ µ. Therefore, it is possible to use Mn as an estimator or approximation of µ. Estimators like Mn can also be used to estimate purely mathematical quantities that are too difficult to compute directly. Such estimators are called Monte Carlo approximations. STA347 - week 12 1

Example Suppose we wish to evaluate the integral This integral cannot easily be solved exactly, but it can be computed approximately using a Monte Carlo approximation. We first note that, I = E(X 2 cos(X 2)) where X ~ Exponential(25). Hence, for large n the integral I is approximately equal to , where , with X1, X2,…i.i.d Exponential(25). Further, in the example on slide 10, we established a method to simulate X ~ Exponential(25). 2 STA347 - week 12

Putting things together we obtain the following algorithm for approximating the integral I. Step 1: Select a large positive integer n. Step 2: Obtain Ui ~ Uniform[0, 1], independently for i = 1, …, n. Step 3: Set , for i = 1, …, n. Step 4: Set , for i = 1, …, n. Step 5: Estimate I by For large n this algorithm will provide a good estimate of the integral I. STA347 - week 12 3

Example Using R The following is the R code for approximating the integral I in the example above. > U = runif(100000) > X = -(1/25)*log((1-U), base = exp(1)) > T = X^2*cos(X^2) > I = mean(T) > I [1] 0.00318922 STA347 - week 12 4

Assessing Error of MC Approximations Any time we approximate or estimate a quantity, we must also indicate how much error there is in the estimate. However, we cannot say what the error is exactly since we are approximating an unknown quantity. Nevertheless, the central limit theorem provide a natural approach to assessing this error, using three times the standard error of the estimate. Thus, we can approximate an unknown quantity such as the integral I in the example above by quoting Mn and the interval STA347 - week 12 5

Assessing Error Using R The following is the R code for assessing the error in the approximation of the integral I. > llimit = I - 3*sd(T)/sqrt(100000) > llimit [1] 0.003121188 > ulimit = I + 3*sd(T)/sqrt(100000) > ulimit [1] 0.003257252   Conclusion: the value of I is approximately 0.003189 and the true value is almost certainly in the interval (0.003121, 0.003257). STA347 - week 12 6

Conditional Probability on a joint discrete distribution Given the joint pmf of X and Y, we want to find and These are the base for defining conditional distributions… STA347 - week 12

Definition For X, Y discrete random variables with joint pmf pX,Y(x,y) and marginal mass function pX(x) and pY(y). If x is a number such that pX(x) > 0, then the conditional pmf of Y given X = x is Is this a valid pmf? Similarly, the conditional pmf of X given Y = y is Note, from the above conditional pmf we get Summing both sides over all possible values of Y we get This is an extremely useful application of the law of total probability. Note: If X, Y are independent random variables then PX|Y(x|y) = PX(x). STA347 - week 12

Example Suppose we roll a fair die; whatever number comes up we toss a coin that many times. What is the distribution of the number of heads? Let X = number of heads, Y = number on die. We know that Want to find pX(x). The conditional probability function of X given Y = y is given by for x = 0, 1, …, y. By the Law of Total Probability we have Possible values of x: 0,1,2,…,6. STA347 - week 12

Conditional densities If X, Y jointly distributed continuous random variables, the conditional density function of Y | X is defined to be if fX(x) > 0 and 0 otherwise. If X, Y are independent then . Also, Integrating both sides over x we get This is a useful application of the law of total probability for the continuous case. STA347 - week 12

Example Consider the joint density Find the conditional density of X given Y and the conditional density of Y given X. STA347 - week 12

Conditional Expectation For X, Y discrete random variables, the conditional expectation of Y given X = x is and the conditional variance of Y given X = x is where these are defined only if the sums converges absolutely. In general, STA347 - week 12

For X, Y continuous random variables, the conditional expectation of Y given X = x is and the conditional variance of Y given X = x is In general, STA347 - week 12

Example Suppose X, Y are continuous random variables with joint density function Find E(X | Y = 2). STA347 - week 12

More on Conditional Expectation Assume that E(Y | X = x) exists for every x in the range of X. Then, E(Y | X ) is a random variable. The expectation of this random variable is E [E(Y | X )] Theorem E [E(Y | X )] = E(Y) This is called the “Law of Total Expectation”. Proof: STA347 - week 12

Example Suppose we roll a fair die; whatever number comes up we toss a coin that many times. What is the expected number of heads? STA347 - week 12

Theorem For random variables X, Y V(Y) = V [E(Y|X)] + E[V(Y|X)] Proof: STA347 - week 12

Example Let X ~ Geometric(p). Given X = x, let Y have conditionally the Binomial(x, p) distribution. Scenario: doing Bernoulli trails with success probability p until 1st success so X : number of trails. Then do x more trails and count the number of success which is Y. Find, E(Y), V(Y). STA347 - week 12