Consider the following cumulative probability distribution.
x 0 1 2 3 4 5
P(X ≤ x) 0.07 0.31 0.46 0.71 0.90 1
a. Calculate P(X ≤ 4). (Round your answer to 2 decimal places.)
b. Calculate P(X = 3). (Round your answer to 2 decimal places.)
c. Calculate P(2 ≤ X ≤ 4). (Round your answer to 2 decimal places.)
a. P(X ≤ 4) = 0.90.
b. P(X = 3) = P(X ≤ 3) - P(X ≤ 2) = 0.71 - 0.46 = 0.25.
c. P(2 ≤ X ≤ 4) = P(X ≤ 4) - P(X ≤ 1) = 0.90 - 0.31 = 0.59.
a. To calculate P(X ≤ 4), we need to look at the value in the cumulative probability distribution table corresponding to x = 4.
From the table, we can see that P(X ≤ 4) = 0.90.
b. To calculate P(X = 3), we can subtract the cumulative probability at x = 2 from the cumulative probability at x = 3.
P(X = 3) = P(X ≤ 3) - P(X ≤ 2)
= 0.71 - 0.46
= 0.25
So, P(X = 3) = 0.25.
c. To calculate P(2 ≤ X ≤ 4), we need to subtract the cumulative probability at x = 1 from the cumulative probability at x = 4.
P(2 ≤ X ≤ 4) = P(X ≤ 4) - P(X ≤ 1)
= 0.90 - 0.07
= 0.83
So, P(2 ≤ X ≤ 4) = 0.83.
To calculate the probabilities mentioned, we can use the information from the cumulative probability distribution given. The cumulative probability at a specific value of X represents the probability that X is less than or equal to that value.
a. To calculate P(X ≤ 4), we can simply look at the cumulative probability for X = 4, which is 0.9. Therefore, P(X ≤ 4) = 0.9.
b. To calculate P(X = 3), we can look at the cumulative probability for X = 3 and subtract the cumulative probability for X = 2. So we have P(X = 3) = P(X ≤ 3) - P(X ≤ 2). Looking at the table, we see that P(X ≤ 3) = 0.71 and P(X ≤ 2) = 0.46. Therefore, P(X = 3) = 0.71 - 0.46 = 0.25.
c. To calculate P(2 ≤ X ≤ 4), we can subtract the cumulative probability for X = 1 from the cumulative probability for X = 4. So we have P(2 ≤ X ≤ 4) = P(X ≤ 4) - P(X ≤ 1). From the table, we find that P(X ≤ 4) = 0.9 and P(X ≤ 1) = 0.07. Therefore, P(2 ≤ X ≤ 4) = 0.9 - 0.07 = 0.83.
Therefore, the answers are:
a. P(X ≤ 4) = 0.9 (rounded to 2 decimal places)
b. P(X = 3) = 0.25 (rounded to 2 decimal places)
c. P(2 ≤ X ≤ 4) = 0.83 (rounded to 2 decimal places)