Answer:
Step-by-step explanation:
To determine the probabilities, we need to calculate the cumulative distribution function (CDF) of the random variable X. The CDF of X gives the probability that X takes on a value less than or equal to a given value. We can calculate the CDF by summing up the probabilities for all values of X less than or equal to the given value.
For the given probability mass function:
x -2 -1 0 1 2
f(x) 1/9 2/9 3/9 2/9 1/9
To calculate F(-2), we need to sum the probabilities for all values less than or equal to -2:
F(-2) = P(X ≤ -2) = f(-2) = 1/9
To calculate F(-1), we need to sum the probabilities for all values less than or equal to -1:
F(-1) = P(X ≤ -1) = f(-2) + f(-1) = 1/9 + 2/9 = 3/9 = 1/3
To calculate F(0), we need to sum the probabilities for all values less than or equal to 0:
F(0) = P(X ≤ 0) = f(-2) + f(-1) + f(0) = 1/9 + 2/9 + 3/9 = 6/9 = 2/3
To calculate F(1), we need to sum the probabilities for all values less than or equal to 1:
F(1) = P(X ≤ 1) = f(-2) + f(-1) + f(0) + f(1) = 1/9 + 2/9 + 3/9 + 2/9 = 8/9
To calculate F(2), we need to sum the probabilities for all values less than or equal to 2:
F(2) = P(X ≤ 2) = f(-2) + f(-1) + f(0) + f(1) + f(2) = 1/9 + 2/9 + 3/9 + 2/9 + 1/9 = 9/9 = 1
Therefore, the exact probabilities are:
F(-2) = P(X ≤ -2) = 1/9
F(-1) = P(X ≤ -1) = 1/3
F(0) = P(X ≤ 0) = 2/3
F(1) = P(X ≤ 1) = 8/9
F(2) = P(X ≤ 2) = 1