Question

A running contest is composed of two consecutive games. The positions of contestants in the second game are determined through the first game results. For that purpose, an urn is prepared to contain balls labeled with contestants’ initials. There is a direct relation between contestants’ rank in the first game and the number of balls labeled with his/her initials. (For instance, if a contestant with initials AC ranked 7th in the first game, there will be 7 balls labeled as AC in the urn). There are 10 participants in the game, and 55 balls in the urn initially. The host will choose 5 balls from the urn with replacement in order to place contestants in the first 5 places. However, at each attempt, a ball with different initials must be chosen (since it is not possible that a contestant starts the game at two different places). If a ball with the same initials is chosen for the second time, it will be discarded and the ball will be put back in the urn. This process will be repeated until a ball with new initials is selected. (As an example, if in the first attempt AC is chosen, then it is not allowed to choose AC again. If a ball with initials AC is selected for the second time, the ball will be put back to the urn and selection will be repeated until something other than AC will be chosen). After the first 5 places are filled, remaining contestants will be placed in inverse order of their rank for the other 5 places (For instance, if the two with the lowest rank among the remaining 5 contestants ranked as 9th and 6th, then contestant ranked as 9 will be put in the sixth place, and the other one will be put in the seventh place). Let X denotes the place of the contestant in the second game who was ranked 9th in the first game.

1-Determine the sample space of the random variable and assign probabilities to the outcomes of X, i.e., get the PMF.

2-Determine the CDF of X.

3-Determine the mean and the variance of X.

4-Determine E[(X − 1)4 ].

Answer 1

1- The sample space of is and the probability for each of the position is .

The probability mass function is  .

2- The CDF is the integration of the probability distribution function that is

3- Mean is thus

and variance is

thus

hence variance is

4- .