Question

6. [R Programming| In order to complete this question, we will need to use a fundamen- tal theorem in probabilty theory known as the (Weak) Law of Large Numbers (WLLN) The WLLN will be covered in greater detail in Chapter 8, however we will use this key result throughout the course to approximate probabilities that are otherwise difficult to calculate theoretically. A simplified version of the theorem goes as follows. Assume we are interested in com- puting the probability of an event A that is otherwise hard to calculate theoretically. We may approximate this probability via a simulation in the following way: . Generate a large number of samples of the data of interest: xi, 22, ..., Tv Let IA(xi)- 0, otherwise For example, in the script Coin Flip Example -R Code.R on Canvas, we want to compute the probability of flipping heads (H) for a coin with unknown probability P(H). We define A -{IH) and we approximate this probability by simulating a large number of coin flips xi є {H,T} for i 1, ,X, and approximating P(H) Σ 1a(zi) The latter is also known as a Simple Monte Carlo Approrimation. Note that the larger the sample size N is, the more accurate the approximation will be. Using this simulation approximation approach for a probability of interest, you are required to approximate the probabilities of interest of the following scenario. Assume that you roll a fair 8-faced die 20 times. You want to approximate the probability that at least i of the rolls have a value of 6 or greater, given that you know at least j of the rolls have a value of 4 or greater, for every combination of i - 12,13,14 and 8,9,10, 11,12 For notational convenience, define the probability of interest by p(i,j) (i.e., we are interested in 3 × 5-15 different probabilities). Your tasks for this question are the following: (a) Write a simple pseudo-code of the steps to approximate p(i,j) for a fixed i and fixed j. (b) Write your own R code to implement your pseudo-code in (a) to approximate all probabilities p(i,j) fori 12,13, 14 and j- 8,9,10, 11,12 via simulation. Note that you must print and staple your R code at the very end of your assignment. (c) Report your numerical values obtained for the approximation of the probabilities p(i, j) for i - 12,13, 14 and j- 8,9,10, 11, 12

Answer 1