Probability Assignments using Statistics Assignment Solution

Discrete Random Variable Assignments

1. The volunteer fire department handles between 0 to 5 calls on any given day. The probability distribution for the number of service calls is

   Number of Calls	Probability
   0	.20
   1	.30
   2	.20
   3	.10
   4	.10
   5	.10

   1. Is this a valid probably assignment? Explain.

      Answer:

      Yes, this is a valid probability assignment, as the sum of probabilities of all events, equals 1.

   2. What is the random variable in this problem?

      Answer:

      The random variable in this problem is the number of calls.

   3. What is the probability that the fire department handles at least 3 service calls on a given day?

      Answer:

      P (Number of calls ≥ 3) = Σ P(X = x), where x varies from 3 to 5 P (Number of calls ≥ 3) = 0.1 + 0.1 + 0.1 = 0.3

   4. What is the probability that the fire department handles at most 2 service calls on a given day?

      Answer:

      P (Number of calls ≤ 2) = Σ P(X = x), where x varies from 0 to 2 P (Number of calls ≤ 2) = 0.2 + 0.3+ 0.2 = 0.

   5. What is the expected number of service calls?

      Answer:

      Expected number of calls = Σx*P(X =x) = 1*0.3 + 2*0.2 + 3*0.1 + 4*0.1 + 5*0.1 = 1.9

   6. What is the variance in the number of service calls?

      Answer:

      Variance = Σx2*p(x) – μ2 Variance = 1*0.3 + 4*0.2 + 9*0.1 + 16*0.1 + 25*0.1 – 1.9*1.9 = 2.49

   7. What is the standard deviation?

      Answer:

      Standard deviation = sqrt (Variance) = sqrt (2.49) = 1.578

      ?
2. Seven plants are operated by a garment manufacturer. They feel there is a ten percent chance for a strike at any one plant and the risk of a strike at one plant is independent of the risk of a strike at another plant.
   1. What is the random variable in this problem?

      Answer:

      Number of strikes on the seven plants.

   2. 1. What is defined as a “success”?

         Answer:

         Here “success” is a strike at a plant.

      2. What is the probability of success?

         Answer:

         P(Success) = 0.1

      3. What is defined as a “failure”?

         Answer:

         Here “failure” is not a strike at a plant.

      4. Calculate the probability of failure.

         Answer:

         P(Failure) = 0.9

   3. Write the probability density function.

      Answer:

      Here X follows a Binomial distribution with n=7 and p=0.1. Thus the probability density function is given by: f(x) = (7¦x) 〖0.1〗^x 〖0.9〗^(7-x), for x=0,1,..,7

   4. What is the probability at most three plants will strike?

      Answer:

      P(at most three plants will strike) = P(X≤3) =P(X=0) + P(X=1) + P(X=2) + P(X=3) =f(0) + f(1) + f(2) + f(3) = 0.4783 + 0.372 + 0.124 + 0.023 = 0.9973

   5. What is the probability that none of the plants will strike?

      Answer:

      P(X=0)= f(0)= 0.4783

   6. What is the probability that two of the plants will strike?

      Answer:

      P(X=2)= f(2)=0.124

   7. What is the probability that at least one of the plants will strike?

      Answer:

      P(X≥1)= 1- P(X=0)= 1-0.4783= 0.5217

   8. What is the probability that between four and six (inclusive) of the plants will strike?

      Answer:

      P(X=4) + P(X=5) + P(X=6) = f(4)+f(5)+f(6) = 0.00255 + 0.00017 + 0.00001= 0.00273

   9. What is the expected number of plants that will strike?

      Answer:

      The expected number of plants that will strike = np = 7*0.1 = 0.7

   10. What is the variance in the number of plants that will strike?

       Answer:

       The variance is given by: np(1-p) 7*0.1*0.9 = 0.63

   11. What is the standard deviation of the number of plants that will strike?

       Answer:

       Standard deviation is the square root of variance = √0.63= 0.7937