Distribution function of a random variable X is given by F(x) = 1 - 1/(x ^ 2); 1 <= x < ∞ Then P(X <= 2) and respectively P(X > 3/2) are
Question
Distribution function of a random variable X is given by F(x) = 1 - 1/(x ^ 2); 1 <= x < ∞ Then P(X <= 2) and respectively P(X > 3/2) are
Solution
The distribution function F(x) is defined as the probability that the random variable X takes on a value less than or equal to x. So, to find P(X <= 2), we simply substitute x = 2 into the distribution function:
F(2) = 1 - 1/(2^2) = 1 - 1/4 = 3/4.
So, P(X <= 2) = 3/4.
To find P(X > 3/2), we need to use the fact that the total probability is 1. The probability that X is greater than 3/2 is the complement of the probability that X is less than or equal to 3/2. So, we find F(3/2) and subtract it from 1:
F(3/2) = 1 - 1/(3/2)^2 = 1 - 4/9 = 5/9.
So, P(X > 3/2) = 1 - 5/9 = 4/9.
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