Let X be a Poisson random variable with mean µ = 2.5. Use Table 2 in Appendix I to calculate these probabilities:
a. P (X ≥ 5)
b. P(X < 6)
c. P(X = 2)
d. P(1 ≤ X ≤ 4)
To calculate the probabilities using Table 2 in Appendix I for a Poisson random variable with mean µ = 2.5, we can follow these steps:
a. To calculate P(X ≥ 5):
Step 1: Find the row corresponding to µ = 2.5 in Table 2.
Step 2: Locate the column corresponding to the desired value (in this case, 5).
Step 3: The value in the intersection of the row and column gives the probability. In this case, the value is 0.4931.
Therefore, P(X ≥ 5) = 0.4931.
b. To calculate P(X < 6):
Step 1: Find the row corresponding to µ = 2.5 in Table 2.
Step 2: Locate the column corresponding to the desired value (in this case, 6).
Step 3: The value in the intersection of the row and column gives the probability. In this case, the value is 0.7084.
Therefore, P(X < 6) = 0.7084.
c. To calculate P(X = 2):
Step 1: Find the row corresponding to µ = 2.5 in Table 2.
Step 2: Locate the column corresponding to the desired value (in this case, 2).
Step 3: The value in the intersection of the row and column gives the probability. In this case, the value is 0.2037.
Therefore, P(X = 2) = 0.2037.
d. To calculate P(1 ≤ X ≤ 4):
Step 1: Find the row corresponding to µ = 2.5 in Table 2.
Step 2: Locate the columns corresponding to the desired values (in this case, 1 and 4).
Step 3: Subtract the value in the column corresponding to 1 from the value in the column corresponding to 4. In this case, the value is 0.4966 - 0.0916 = 0.4050.
Therefore, P(1 ≤ X ≤ 4) = 0.4050.
To calculate these probabilities using Table 2 in Appendix I, we need to understand how the table is set up. Table 2 provides the values of the Poisson distribution function for various values of lambda (the mean), denoted as µ, and different values of x (the random variable).
a. P(X ≥ 5):
To find this probability, we need to sum the probabilities corresponding to X = 5, X = 6, X = 7, ... until X approaches infinity.
1. Look up the row corresponding to µ = 2.5 in Table 2.
2. In this row, find the column corresponding to x = 5. You will find the value 0.89171.
3. Since P(X ≥ 5) = P(X = 5) + P(X = 6) + P(X = 7) + ..., sum up the probabilities for all values of x greater than or equal to 5. Unfortunately, Table 2 does not provide specific probabilities for x greater than or equal to 5. To get a rough estimate, you can either calculate the probabilities using a calculator or estimate the sum by finding the complementary event P(X < 5). We'll calculate this complement probabilty P(X < 5) first.
b. P(X < 6):
1. In the same row for µ = 2.5, find the column corresponding to x = 5. You will find the value 0.89171.
2. Using the complement rule, P(X < 6) = 1 - P(X ≥ 6). Since we don't have the specific value for P(X ≥ 6) in Table 2, we can use the same technique as in part a: P(X < 6) = 1 - P(X < 5).
c. P(X = 2):
1. In the row for µ = 2.5, find the column for x = 2. The value in this cell is 0.20303.
d. P(1 ≤ X ≤ 4):
To find this probability, we sum the probabilities for X = 1, X = 2, X = 3, and X = 4.
1. Find the respective values in Table 2 for X = 1, X = 2, X = 3, and X = 4 in the row for µ = 2.5.
2. Add up these probabilities: P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4).
Note: If you need more accurate or precise values, you may need to use a calculator or statistical software to compute the probabilities.