Example 3.12 Starting at a fixed time, we observe the gender of each newborn chi

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Example 3.12 Starting at a fixed time, we observe the gender of each newborn child at a certain hospital until a boy (8) is born. Let p·P(B), assume that successive births are independent, and define the rv X by xnumber of births observed. Then and Continuing in this way, a general formula emerges: p)-Pp- 1, 2, 3, otherwise and The expression The parameter p can assume any value between above describes the family of geometric distributions. In the gender example, p = 0.51 might be appropriate, but if we were looking for the first child with Rh-positive blood, then we might have p = 0.85. 4 Question Details DevoreStat9 3.AE.017. [3232880]. Example 3.17 Just after birth, each newborn child is rated on a scale called the Apgar scale. The possible ratings are 10, with the child's rating determined by color, muscle tone, respiratory effort, heartbeat, and reflex irritability (the best possible score is 10) Let X be the Apgar score of a randomly selected child born at a certain hospital during the next year, and suppose that the pmf of X is 3 4 5 67 8 9 10 p(x) 0.001 0.002 0.002 0.005 0.02 0.04 0.14 0.43 0.24 0.1 0.02 Then the …Select- value of X is E(X)- = 0( +1(0.002) +2(0.002) +8(O.24) 9(0.1)10(0.02) (entered to three decimal places) Again, is not a possible value of the variable X. Also, because the variable refers to a future child there is no concrete existing population to which refers. Instead, we think of the pmf as a model for a conceptual population population is then of the values 0, 1, 2,.10 10. The mean value of this conceptual (entered to three decimal places). S. Question Details DevoreStat9 3.AE.023. (3232913 Example 3.23 A computer store has purchased three computers of a certain type at $500 apiece. It will sell them for $1000 apiece. The manufacturer has agreed to repurchase any computers still unsold after a specified period at $200 apiece. Let X denote the number of computers sold, and suppose that pO) 0.1, D( 1 ) = 0.2, p(2)s 0,3, and p(3)s X units, the given information that h(x) - revenue- cost - 1000x+200(3 X) 1500 800X-900. The expected profit is then With h(Xx) denoting the profit associated with selling Eh(X)) h(0) p(0)(1) p(1) (2) p(2) +h(3) p(3) (-900)(0.1) + ( 00.2) + (700)(0.3) + ( (0.4)

Explanation / Answer

Example: 3.12 Probability that the gender of a newborn baby is boy is P(B)=0.5

Hence,

P(1)= P(X=1) = P(B)= 0.5

P(2)=P(GB)=P(G)*P(B)=0.5*0.5=0.25

P(3)=P(GGB)=P(G)*P(G)*P(B)=0.5*0.50.5=0.125

The Parameter P can assume any value between 0 and 1.

Example 3.17.

The Expected value of X is given by

E(X) = mu = 0.*(0.001) +1*(0.002) +2*(0.002) +3*(0.005) +5*(0.04) +6*(0.14)+ 7*(0.43) +8*(0.24) +9*(0.1) +10*(0.02)

That is E(X) = 7.091

Example: 3.23

Here P(3)= 1- (P(0) +P(1) + P(2))

= 1- (0.1+0.2 +0.3)

P(3) =0.4

Here h(1)= 800-900=-100

h(3)= 800*3 -900 =1500

Hence

E(h(x)) =h(0)*p(0) +h(1)*p(1) +h(2)*p(2) +h(3)*p(3)

= -900*(0.1) -100*(0.2) + 700*(0.3) + 1500*(0.4)

= -90 - 20 +210 + 600

=700

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