We consider a six-sided die, which is slightly unbalanced: each of the numbers 1, 2, 3, 4, and 5 comes lip with probability p = 0.16, while 6 comes up with probability q = 0.20. We throw the die once, and we denote by X [E {1, 2, 3, 4, 5,6" the number obtained. (i) (6 points) Compute the expectation ]E[X] of X. (ii) (6 points) Compute the variance Var{X) of X. We now throw the die repeatedly 10000 times, and we denote by X1, . . . , X10000 the numbers obtained. Let 81mm = X1 + . .. + X10030 be the total sum of the 10000 throws (which we assmne to be independent). (iii) (8 points) Compute iE[Slnmg] and Var(81mm). (iv) (10 points) Using the Central Limit Theorem, give an approximation of P0910000 2 37000) in terms of the cumulative distribution function ¢u(:r) = %f e't2/2dt of a random variable with standard normal distribution (i.e. NH], 1)).