Consider the following joint discrete probability mass function (pmf) p(x, y)=0.

Question

Consider the following joint discrete probability mass function (pmf) p(x, y)=0.1 for (x, y)= (1, 1), (2, 1), (1, 2), and (2, 2) 0.2 for (x, y) =(0, 0) 0.4 for (x, y)=(3, 3) 1.1. obtain conditional pmf of X when Y= 1. 1.2. Clearly showing all of your calculations, calculate the correlation coefficient for X and Y. The number of flaws in a glass sheet has a mean value of 0.5 and a standard deviation of 0.7071. What is the probability that the total number of flaws in 100 sheets of glass is less than 40?

Explanation / Answer

1.1 Here we have to obtain conditional pmf of X when Y =1

so there are only two values of X when Y =1, these are X =1 and X =2

all possible values of X are 0,1,2 and 3

so P(0) = 0/ 0.2 = 0

P(1) = 0.1/(0.1+0.1) = 0.5

P(2) = 0.1/( 0.1+ 0.1) = 0.5

P(3) = 0/ 0.2 = 0

Q 1.2 The correaltion coefficient of X and Y isXY

XY=Corr(X,Y)=Cov(X,Y)/XY=XY/XY

We will calculate X first by x

fx(x) = 0.2; x = 0

= 0.2; x =1

   = 0.2 ; x = 2

   = 0.4 ; x = 3

x = 0.2 * 0 + 0.2 * 1 + 0.2 * 2 + 0.4 * 3 = 1.8

X2 = 0.2 * 1.82 + 0.2 * 0.82 + 0.2 * 0.22 + 0.4 * 1.22 = 1.36

X = 1.166

We will calculate y first by y

fx(x) = 0.2; y = 0

= 0.2; y =1

   = 0.2 ; y= 2

   = 0.4 ; y = 3

y = 0.2 * 0 + 0.2 * 1 + 0.2 * 2 + 0.4 * 3 = 1.8

y2 = 0.2 * 1.82 + 0.2 * 0.82 + 0.2 * 0.22 + 0.4 * 1.22 = 1.36

y = 1.166

Now we will calculate XY

2XY = 0.4 * ( 3 - 1.8)* ( 3-1.8) + 0.2 *( 0 -1.8) * ( 0 - 1.8) + 0.1 * ( 1- 1.8) * ( 1- 1.8) + 0.1 * ( 2 -1.8) * ( 1 -1.8) + 0.1 * ( 1 - 1.8) * ( 2 - 1.8) + 0.1 * ( 2 - 1.8) * ( 2 - 1.8) = 1.26

XY = 1.225

so correlation coefficient = XY/XY = 1.225 / 1.36 = 0.9

Q.2 Mean value = 0.5 and standard deviation = 0.7071

so Here we have to calculate that Total number of flaws in 100 sheets of glass is less than 40

so on an average number of faults on the sheet = 100 * 0.5 = 50

and standerd deviation = n * [/ n] = 100 * [ 0.7071/ 100] = 7.071

so P( X< 40 ; 50; 7.071) by normal distribution

Z - value = ( 40- 50) / (7.071) = - 1.4142

so P( X< 40 ; 50; 7.071) = 0.0785

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