Answer:
a
[tex]P(K = k ) = \ ^nC_x * p^k * (1-p)^{n-k}[/tex]
Here k is the random number of number of pairs of shoes that will be returned for refund, which can be k = 0,1 ,2,3,4 ..., 120
b
The mean is [tex]E(K) = np = 13.2[/tex]
The standard deviation is [tex]\sigma = 3.43[/tex]
c
The mean is [tex]E(C) = E(105 K) = 1386[/tex]
The standard deviation is [tex]\sigma = 381.48 [/tex]
Step-by-step explanation:
From the question we are told that
The cost to dealer for each refund is [tex]C = \$105[/tex]
The probability that a shoe will refunded is [tex]P(x) = 0.11[/tex]
The number of shoes purchased is n = 120 pairs
Generally the probability distribution of the number of pairs of shoes that will be returned for refunds is
[tex]P(K = k ) = \ ^nC_x * p^k * (1-p)^{n-k}[/tex]
Here k is the random number of number of pairs of shoes that will be returned for refund, which can be k = 0,1 ,2,3,4 ..., 120
Generally the mean is mathematically represented as
[tex]E(K) = np = 120 * 0.11[/tex]
=> [tex]E(K) = np = 13.2[/tex]
Generally the standard deviation is mathematically represented as
[tex]\sigma = \sqrt{ n* p (1-p)}[/tex]
=> [tex]\sigma = \sqrt{ 120* 0.11 (1-0.11)}[/tex]
=> [tex]\sigma = 3.43[/tex]
Generally the total refund cost is mathematically represented as
[tex]C = 105 K[/tex]
Here K denotes the number of shoes returned
Generally the mean of the total refund cost is mathematically represented as
[tex]E(C) = E(105 K) = 105 E(K)[/tex]
=> [tex]E(C) = E(105 K) = 105 * 13.2[/tex]
=> [tex]E(C) = E(105 K) = 1386[/tex]
Generally the variance of the total refund cost is mathematically represented as
[tex]Var(K) = 105^2 * E(K)[/tex]
=> [tex]Var(K) = 105^2 * 13.2[/tex]
=> [tex]Var(K) = 145530[/tex]
=> [tex]\sigma = \sqrt{145530 }[/tex]
=> [tex]\sigma = 381.48 [/tex]
