Question
Download Solution PDFLet X be a random variable with cumulative distribution function given by Then the value of is equal to
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcepts Used:
1. Cumulative Distribution Function (CDF):
The CDF F(x) gives the probability that the random variable X takes a value less than or equal to x . That is, F(x) = P(X ≤ x) .
2. Finding Probability Using the CDF:
The probability that the random variable X lies within a certain interval (a, b] is given by:
P(a < X ≤ b) = F(b) - F(a)
3. Probability at a Specific Point (Jump Discontinuity):
The probability at a specific point x = c is the difference in the CDF just to the right and just to the left of c :
P(X = c) = F(c+) - F(c-)
Explanation -
We are given a cumulative distribution function (CDF) F(x) of a random variable X as:
F(x) =
From the definition of the probability from the CDF: P(a < X
In this case, we need to calculate
Since
Thus, the probability
=
The probability at a point is the jump in the CDF at that point. We need to calculate F(0+) - F(0-) .
From the CDF definition: F(0+) = F(0) =
⇒ F(0-) = 0
Thus,
Now, we add the two results:
Thus, the final answer is 17/36.
Last updated on Jun 5, 2025
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