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A restaurant needs a staff of 3 waiters and 2 chefs to be properly staffed. The joint probability model for the number of waiters (X) and chefs (Y) that show up on any given day is given below. X Y 0 1 2 3 0 k 0.01 0.01 0.03 1 0.02 0.04 0.04 0.06 2 0.02 0.01 0.06 0.63 a) What must the value of k be for this to be a valid probability model

Respuesta :

Solution :

Given the Joint Probability distribution is :

Y|X     0         1           2           3            P(y)

0       k       0.03    0.03      0.03      k+0.09

1      0.01    0.02    0.04      0.04        0.11

2     0.01    0.01     0.05      0.72        0.79

P(x) k+0.02  0.06   0.12      0.79       k+0.99

 

Since we know,

[tex]$\sum \sum P(x_i,y_j) = 1$[/tex]

⇒ k + 0.99 = 1

⇒ k = 1 - 0.99

⇒ k = 0.01

Therefore, the value of k is 0.01