Consider the following cumulative probability distribution.
x 0 1 2 3 4 5
P(X ≤ x) 0.11 0.32 0.45 0.75 0.83 1
a. Calculate P(X ≤ 3). (Round your answer to 2 decimal places.)
b. Calculate P(X = 2). (Round your answer to 2 decimal places.)
c. Calculate P(1 ≤ X ≤ 3). (Round your answer to 2 decimal places.)
a. P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.11 + 0.32 + 0.45 + 0.75 = 1.63. Rounded to 2 decimal places, P(X ≤ 3) is 1.63.
b. P(X = 2) = P(X ≤ 2) - P(X ≤ 1) = 0.45 - 0.32 = 0.13. Rounded to 2 decimal places, P(X = 2) is 0.13.
c. P(1 ≤ X ≤ 3) = P(X = 1) + P(X = 2) + P(X = 3) = 0.32 + 0.45 + 0.75 = 1.52. Rounded to 2 decimal places, P(1 ≤ X ≤ 3) is 1.52.
To calculate the probabilities, we can use the cumulative probability distribution given.
a. P(X ≤ 3) means the probability of X being less than or equal to 3. Looking at the table, we can see that the cumulative probability for X ≤ 3 is 0.75. Thus, P(X ≤ 3) = 0.75.
b. P(X = 2) means the probability of X being exactly equal to 2. To calculate this, we need to find the difference between the cumulative probabilities at 2 and 1. Looking at the table, we can see that the cumulative probability for X ≤ 2 is 0.45, and the cumulative probability for X ≤ 1 is 0.32. Hence, P(X = 2) = P(X ≤ 2) - P(X ≤ 1) = 0.45 - 0.32 = 0.13.
c. P(1 ≤ X ≤ 3) means the probability of X being between 1 and 3, inclusive. To find this, we need to subtract the cumulative probability at 1 from the cumulative probability at 3. Looking at the table, we can see that the cumulative probability for X ≤ 3 is 0.75, and the cumulative probability for X ≤ 1 is 0.32. Therefore, P(1 ≤ X ≤ 3) = P(X ≤ 3) - P(X ≤ 1) = 0.75 - 0.32 = 0.43.
To answer these questions, we need to understand the cumulative probability distribution function. The cumulative probability distribution function (CDF) gives the probability that a random variable takes a value less than or equal to a given value.
a) To calculate P(X ≤ 3), we need to find the cumulative probability for x = 3. According to the given distribution, P(X ≤ 3) is 0.75. Therefore, P(X ≤ 3) = 0.75.
b) To calculate P(X = 2), we need to find the probability of the random variable taking the specific value of 2. We can find this by subtracting the cumulative probability of the previous value from the cumulative probability of the desired value. In this case, P(X = 2) is (cumulative probability at x = 2) - (cumulative probability at x = 1). According to the given distribution, P(X ≤ 2) is 0.45 and P(X ≤ 1) is 0.32. Therefore, P(X = 2) = 0.45 - 0.32 = 0.13.
c) To calculate P(1 ≤ X ≤ 3), we need to find the cumulative probability for x = 3 and subtract the cumulative probability for x = 0. In this case, P(1 ≤ X ≤ 3) is (cumulative probability at x = 3) - (cumulative probability at x = 0). According to the given distribution, P(X ≤ 3) is 0.75, and P(X ≤ 0) is 0.11. Therefore, P(1 ≤ X ≤ 3) = 0.75 - 0.11 = 0.64.
Therefore, the answers to the questions are:
a) P(X ≤ 3) = 0.75
b) P(X = 2) = 0.13
c) P(1 ≤ X ≤ 3) = 0.64.