Discuss the below:
Q1. Place eight chips in a bowl: Three have the number 1 on them, two have the number 2, and three have the number 3. Say each chip has a probability of 1/8 of being drawn at random. Let the random variable X equal the number on the chip that is selected, so that the space of X is S = {1, 2, 3}. Make reasonable probability assignments to each of these three outcomes, and computer the mean μ and the variance σ^2 of this probability distribution.
Q2. A fair coin is flipped successively at random until the first head is observed. Let the random variable X denote the number of flips of the coin that are required. Then the space of X is S = {x : x = 1, 2, 3, 4,...}. Later we learn that, under certain conditions, we can assign probabilities to these outcomes in S with the function f(x) = (1/2)^x, x = 1, 2, 3, 4,... Compute the mean for μ. Hint: Write out the series for μ, and then construct the series for (1/2)μ and take the difference. An alternative method would be to compare the series for μ with that of the negative binomial (1 - z)^-2, with z = 1/2.
Q3. Let X equal the number of calls per hour received by 911 between midnight and noon and reported in the Holland Sentinel. On October 29 and October 30, the following numbers of calls were reported:
October 29: 0, 1, 1, 1, 0, 1, 2, 1, 4, 1, 2, 3
October 30: 0, 3, 0, 1, 0, 1, 1, 2, 3, 0, 2, 2