3. A manufacturing company measures the length and the width of a rectangular bo

Question

3. A manufacturing company measures the length and the width of a rectangular box that it produces. The measurements are rounded to the nearest cm (so they are discrete). Let X denote the length and Y denote the width. The possible values of X are 129, 130, and 131 cm and, for Y are 15 and 16 cm. Thus there are 6 possible pairs. The following table gives the probability for each pair: x 129 x = 130 x-131 y = 15 0.12 0.42 0.06 y= 16 0.08 0.28 0.04 a. Find the probability that the length is 129cm? b. Find the marginal distribution of X and c. Find the probability that the box has a length of 130cm given the width is 15cm d. Find E[Y IX-129] and Var(Y|X-129).

Explanation / Answer

a)from above probability that length isd 129 cm =P(X=129)=P(X=129,Y=15)+P(X=129,Y=16)=0.12+0.08=0.20

b)

from above marginal distribution of X is as follows:

and  marginal distribution of Y is as follows:

c)

here probability =P(X=130|Y=15)=P(X=130,Y=15)/P(Y=15) =0.42/0.60 = 0.70  

d) here condition distribution of Y given X :

from above:

E(Y|X=129)= 15.4

and Var(Y|X=129) = E(Y2|X=129)-(E(Y|X=129))2 =237.4-15.42=0.24

please revert for any clarificaiton requiired

x P(x) 129 0.2000 130 0.7000 131 0.1000

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