Problem 1: A research group consists of 3 biology students and 4 chemistry students. We are selecting 2 students at random (without replacement) from this group of 7 students. The random variable X is the number of biology students among the students selected during our experiment.
(a) What are the possible values of X?
(b) For each possible value of X, determine its probability.
(c) Present the distribution law (the probability mass function) of the random variable X as a table.
(d) Depict the probability mass function of X as a bar graph.
(e) Calculate the mathematical expectation of X.
(f) Calculate the variance of X and the standard deviation of X.
(g) Calculate the cumulative distribution function of X.
Problem 2: Assume the probability that a newborn is a boy is 50 %, that is, p = 0.5. Suppose a family has 8 children.
(a) What is the distribution law followed by the number of boys in such a family?
(b) Denote F the cumulative distribution function of this distribution law. Express each of the following probabilities through the values of F: i. there are at least 4 boys, ii. there are at least 4 girls, iii. all children are boys, iv. there are exactly 5 boys and 3 girls, v. there are between 3 and 5 boys.
(c) Using R, calculate all of the above values. Attach the printout of your R workspace.