Question

Name the distribution which seems most appropriate to each of the following random variables and specify the values of the associated parameters. [Example: “The number of students in a class of size 42 who pass Mech. Eng. 1234, given that, on average, the proportion of students who pass Mech. Eng. 1234 is 0.6”. Answer: Binomial; n = 42, p = 0.6.]

1. (i) The number of digits generated randomly and independently from {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} to obtain the first occurrence of “3”.

2. (ii) The number of computers, in a lab containing 20 computers, which fail before their warranty expires, given that 5% of such computers fail before their warranty expires.

3. (iii) The number of Dell computers in a lab containing 20 computers which were chosen at random from a supply of 100 computers, 5 of which were Dells.

4. (iv) The number of reflected sub-atomic particles in an evacuated duct of a nuclear fusion reactor, when 50 particles are released in the duct. For this particular duct, 16% of all such particles are reflected, and 84% of all such particles are absorbed. The particles behave independently of each other.

5. (v) The number of graduate students on a committee of size 5 which is chosen at random from a university department consisting of 15 faculty members and 23 graduate students.

6. (vi) The number of households sampled by a sociologist, who samples until he obtains a house- hold whose head is a single female parent, given that 12% of all households are headed by a single female parent.

7. (vii) The number of deer caught and inspected by Wildlife Officers, up to the time when they catch a tagged deer, given that 1.5% of all deer are tagged.

Answer 1

i. Geometric distribution with parameter p =1/10

The geometric distribution is useful to model the number of failures before the first success. Here getting 3 is the success.

ii. Binomial distribution with parameter n=20 , p= 0.05

The binomial distribution model allows us to compute the probability of observing a specified number of "successes" when the process is repeated a specific number of times and the outcome for a given trail is either a success or a failure.

iii. Hypergeometric distribution with parameter N = 100, n=20, M= 5

hypergeometric distribution is a discrete probability distribution that describes the probability of x successes (random draws for which the object drawn has a specified feature) in draws, without replacement, from a finite population of size that contains exactly M objects with that feature, wherein each draw is either a success or a failure.

iv. Binomial distribution with parameter n= 50, p= 0.16