A fair die is thrown twelve times. Determine the expected sum of eyes and the variance of the sum of eyes.

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Always give a complete calculation method and a comprehensible justification for your solutions. For tasks with R, the input and the output together with the explanation must be submitted.

Task 15: (2 points) A fair die is thrown twelve times. Determine the expected sum of eyes and the variance of the sum of eyes.

Exercise 16: (4 points) A Bernoulli experiment with success probability p is repeated independently.

(a) Let p = 25%. What is the probability of the second success being observed on the fourth iteration? What is the probability of the third success being observed on the fourth iteration?

(b) What is the probability of observing the kth success at the nth repetition?

 

Exercise 17: (5 points) Consider the density of exercise 13 again:

f : R → [0, ∞), f (x) = 1 x1[2,4](x).

There F (t) = 1 (t2 4) was calculated in the case t [2, 4] and thus F−1(s) = √12s + 4 for s (0, 1). With this, the R function create can be implemented as follows: create <- function ( n) sqrt (12 * runif ( n )+4)

(a) Use this to generate n = 1 000 000 pseudo-random numbers according to the distribution with the above density. Use R to determine the mean, empirical variance, and histogram of these pseudo-random numbers. What is noticeable and why?

(b) Calculate 10 000 times the mean of always new n = 10 000 pseudo-random numbers according to the distribution with the above density. Create a histogram with breaks=50 for these mean values. What do you notice? Note: replicate.

(c) Let X1, X2, . . . independent and identically distributed random variables with the above

Density. Determine the expectation and variance of

1 (X1 + • • • + Xn)

in the case n = 10 000.

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