Question

2. The probability that a person in their 20 s (age 20−29 ) living in the Greater Toronto Area moves to a new apartment in any given year is 0.16. (30 points total) a. What is the theoretical mean and theoretical variance of this geometric distribution? b. What is the probability that a twenty-something will make their first move of their 20 s when they are 22 years old? c. What is the probability that a twenty something will live in the same place for their entire twenties? d. What is the probability that a twenty-something will move for the first time during their twenties when they are of age 24 to age 26 ? e. What is the probability that a twenty-something will NOT move for the first time during their twenties when they are of age 20 to age 22 ? f. Diagram the geometric distribution that represents the probability a twenty-something will move apartments from age 20 to 29.

Answer 1

a) The theoretical mean and theoretical variance of this geometric distribution are 6.25 and 32.81, respectively.

b) The probability that a twenty-something will make their first move when they are 22 years old is 0.113.

c) The probability that a twenty-something will live in the same place for their entire twenties is 0.0175.

d) The probability that a twenty-something will move for the first time during their twenties when they are age 24 to age 26 is 0.20278.

e) The probability that a twenty-something will NOT move for the first time during their twenties when they are of age 20 to age 22 is 0.5927.

f) The diagram of the geometric distribution that represents the probability a twenty-something will move apartments from age 20 to 29 looks like a series of bars.

The x-axis represents the age at which the person makes their first move, and the y-axis represents the probability of making a move at that age.

The bars would start at age 20 and continue to age 29, with the highest bar being at age 20 (representing the probability of making a first move at that age), and the bars getting progressively lower as the age increases.

How the probabilities are determined:

The probability that a person in their 20s (age 20−29 ) living in the Greater Toronto Area moves to a new apartment in any given year as p = 0.16

a. The theoretical mean (expected value) and variance of a geometric distribution are given by:

Mean: E(X) =
`(1)/(p)` =
`(1)/(0.16)` ​≈ 6.25

Variance: Var(X) =
`(1 - p)/(p^2)`

=
`(1 - 0.16)/(0.16^2)`

=
`(0.84)/(0.0256)`

= 32.8125

= 32.81

b. The probability that a twenty-something will make their first move when they are 22 years old is the probability that they don’t move for 2 years and then move in the third year. This is given by:

P(X=3) =
`(1 - p)^(3- 1) * p`

= (1 - 0.16)² x 0.16

= 0.84² x 016

= 0.112896

= 0.113

c. The probability that a twenty-something will live in the same place for their entire twenties is the probability that they don’t move for 10 years. This is given by:

P(X>10) =
`(1 - p)^{10`

=
`(1 - 0.16)^{10`

= 0.84¹⁰

= 0.1749

≈ 0.0175

d. The probability that a twenty-something will move for the first time during their twenties when they are of age 24 to age 26 is the sum of the probabilities that they move for the first time at age 24, 25, or 26. This is given by:

P(5 ≤ X ≤ 7)

= P(X=5) + P(X=6) + P(X=7)

= [(1−p)^4 x p] + [(1−p)^5 x p] + [(1−p)^6 x p]

= [(1−0.16)^4 x 0.16] + [(1−0.16)^5 x 0.16] + [(1−0.16)^6 x 0.16]

= [(0.84)^4 x 0.16] + [(0.84)^5 x 0.16] + [(0.84)^6 x 0.16]

= [0.49787 x 0.16] + [0.41821 x 0.16] + [0.351298 x 0.16]

= 0.0796592 + 0.0669136 + 0.05620768

= 0.20278

e. The probability that a twenty-something will NOT move for the first time during their twenties when they are of age 20 to age 22 is the probability that they don’t move for 3 years. This is given by:

P(X>3)

= (1−p)³

= (1−0.16)³

= 0.84³

= 0.5927