G4. you start a dog walking business. define to be a random variable denoting how many dogs you walkthis week. the probability mass function (pmf),f(x), o'is defined as follows:
0 1 2 3 4 5 6 7 p(x = x) 0.14 0.12 0.15 0.23 0.18 0.09 0.08 0.01
(a) verify that f(x) is a valid probability mass function.
(b) find the probability that you will walk at least two dogs this week.
(c) compute the expected number of dogs you will walk this week.
(d) compute the expected value of x2.
(e) compute var[x] = e(x2) – (e[x])2.

(a) Yes, f(x) is a valid probability mass function.

(b) The probability that you will walk at least two dogs this week = 0.74.

(c)  The expected number of dogs you will walk this week = 3 dogs.

(d) The expected value of X2 = 11.29

(e) Var[x] = E(X2) – (E[X])2 = 3.2811

Step-by-step explanation:

We are given with the probability mass function (pmf),f(x), o'is defined as follows:

Firstly let X = Number of dogs you walk this week

    X                 P(X = x)

    0                      0.14

    1                       0.12

    2                      0.15

    3                      0.23

    4                      0.18

    5                      0.09

    6                      0.08

    7                       0.01

(a) Now f(x) to be a valid probability mass function, two conditions should be    met :

So, First condition is already met as all values are positive and for second condition = 0.14 + 0.12+ 0.15+ 0.23+ 0.18+ 0.09+ 0.08+ 0.01 = 1

Hence both the conditions are satisfied so f(x) is a valid probability mass function.

(b) Probability that we will walk at least two dogs this week = P(X>=2)

      = 1 - P(X = 0) - P(X = 1) = 1 -  0.14 - 0.12 = 0.74

(c) To Compute the expected number of dogs you will walk this week we will use expectation formula which says:

           E(X) = =

                                           = 2.83 or 3 after rounding off.

Therefore the expected number of dogs you will walk this week are 3 dogs.

(d) The expected value of X2 [E(X2)] =  

      = = 11.29

(e) Var[x] = E(X2) – (E[X])2

     We have E(X2) = 11.29 and E(X) = 2.83

      Var[x] = = 3.2811.

300

step-by-step explanation:

28-600w < 300

28-600w > 300

600-28w < 300

600-28w > 300