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. The sandwich shop offers 8 different sandwiches. Jamey likes them all equally. He picks one randomly each day for lunch. During a given week of 5 days, let X be the number of times he chooses salami, Y the number of times he chooses falafel, and Z the number of times he chooses veggie. Find the joint probability mass function of (X, Y, Z). Identify the distribution by name.

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ajeigbeibraheem

Answer:

Step-by-step explanation:

From the given information:

A sandwich shop offers eight types of sandwiches, and Jamey likes all of them equally.

The probability that Jamey picks any one of them is 1/8

Suppose

X represents the number of times he chooses salami

Y represents the number of times he chooses falafel

Z represents the number of times he chooses veggie

Then  X+Y+Z ≤ 5 and;

5-X-Y-Z represents the no. of time he chooses the remaining

8 - 3 = 5 sandwiches

However, the objective is to determine the P[X=x,Y=y,Z=z] such that 0≤x,y,z≤5

So, since he chooses x no. of salami sandwiches with probability (1/8)x

and y number of falafel with probability (1/8)y

and for z (1/8)z

Therefore, the remaining sandwiches are chosen with probability [tex]\dfrac{5}{8} (5-x-y-z)[/tex]

So. these x days, y days and z days can be arranged within five days in

[tex]= \dfrac{5!}{x!y!z!(5-x-y-z)!}[/tex]

Thus;

[tex]P[X=x,Y=y,Z=z]= \dfrac{5!}{x!y!z!(5-x-y-z)} \times \dfrac{1}{8}x*\dfrac{1}{8}y* \dfrac{1}{8}z* \dfrac{5}{8}(5-x-y-z)[/tex]

since 0 ≤ x, y, z ≤ 5 and x + y + z ≤ 5.

The distribution is said to be Multinomial distribution.

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