In the following table, the random variable x represents the number of students

Question

In the following table, the random variable x represents the number of students that a statistics tutor may see on any given day and P(x) represents the probability that the tutor sees that number of students. x 0 1 2 3 4 5 P(x) 0.11 0.18 0.36 0.16 0.14 0.05

a) Confirm if this is a legitimate probability distribution by stating the conditions that must be satisfied and describing/showing how they are satisfied. Explain.

b) Find and report the mean and the standard deviation of this distribution.

c) Based on this distribution, what is the probability that a tutor sees at least two students on a certain day?

d) Based on this distribution, what is the probability that a tutor sees either three or four students on a certain day?

e) Based on this distribution, what is the probability that a tutor sees exactly 10 students on a certain day?

Explanation / Answer

a)

Yes, the conditions are satisfied, as the probabilities sum up to 1 (0.11+0.18+0.36+0.16+0.14+0.05=1), and all probabilities are between 0 and 1.

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b)

Consider:

Thus,  
  
E(x) = mean = Sum(xP(x)) =    2.19 [ANSWER]

Also,

Var(x) = E(x^2) - E(x)^2 =    1.7539

Thus,

s(x) = sqrt [Var(x)] =    1.324348897 [ANSWER, STANDARD DEVIATION]

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c)

Hence,

P(at least 2) = 1 - P(0) - P(1)

= 1 - 0.11 - 0.18

= 0.71 [ANSWER]

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d)

Also,

P(3 or 4) = P(3) + P(4) = 0.16 + 0.14 = 0.30 [ANSWER]

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e)

From the table, the maximum value of x is 5, so 10 is impossible,

P(10) = 0 [ANSWER]

x P(x) x P(x) x^2 P(x) 0 0.11 0 0 1 0.18 0.18 0.18 2 0.36 0.72 1.44 3 0.16 0.48 1.44 4 0.14 0.56 2.24 5 0.05 0.25 1.25

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