During the 2016 season, the average length of a Houston Astros baseball game was

Question

During the 2016 season, the average length of a Houston Astros baseball game was 3 hours, 4 minutes, with a standard deviation of 29 minutes. Assume that the dis- tribution of these game lengths is normal. If a game from that season is selected at random, find the probability that the length of the game is. . .

(a) . . . more than 3 hours, 15 minutes.
(b) . . . between 2 hours, 30 minutes and 2 hours, 45 minutes.

(c) . . . less than 2 hours, 15 minutes.

Let X be a random variable with mean 2.5 and standard deviation 0.9, and let Y be a random variable with mean 3.1 and standard deviation 1.4. Assume X and Y are independent.

(a) Find E[X + 2Y ]. (b) Find E[X Y +1].

(c) Find Var(2X).
(d) Find Var(2X Y ).

(e) Find E[X2].

Explanation / Answer

here mean =3 hour 4 minute =184 minute and std deviaiton=29 minutes

a) probability that the length of the game is  more than 3 hours, 15 minutes =P(X>195)=P(Z>(195-184)/29)

=P(Z>0.3793)=0.3522

b). between 2 hours, 30 minutes and 2 hours, 45 minutes =P(150<X<165)=P(-1.1724<Z<-0.6552)

=0.2562-0.1205 =.1357

c)less than 2 hours, 15 minutes =P(X<135)=P(Z<-1.6897)=0.0455

2)

a)E(X+2Y)=E(X)+2E(Y)=2.5+2*3.1=8.7

b)E[X Y +1] =E(X)-E(Y)+1 =2.5-3.1+1 =0.4

c) Var(2X) =4*Var(X) =4*(0.9)2 =3.24

d) Var(2X-Y)=4*Var(X) +Var(Y) =4*0.92+1.42 =5.2

e)E(X2) =Var(X)+(E(X))2 =0.92+2.52 =7.06

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