40. For a discrete random variable X, ran(X) = {0, 1, 2, 3} and the cumulative probability
F(X) is shown below:
| X | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| F(X) | 0.5 | 0.6 | 0.8 | 1.0 |
The mean value of X is __________.
Introduction
Finding the mean of a discrete random variable from cumulative probability
is a common problem in statistics for exams such as GATE, CSIR-NET, JAM, and university
examinations. When the cumulative distribution function (CDF) is given, the probability
mass function (PMF) must first be obtained before calculating the mean.
Step 1: Understanding the CDF
The cumulative distribution function is defined as:
F(x) = P(X ≤ x)
To calculate the mean, we must first determine the probability mass function (PMF):
P(X = x) = F(x) − F(x−)
Step 2: Deriving the Probability Mass Function
| X | F(X) | P(X = x) |
|---|---|---|
| 0 | 0.5 | 0.5 |
| 1 | 0.6 | 0.6 − 0.5 = 0.1 |
| 2 | 0.8 | 0.8 − 0.6 = 0.2 |
| 3 | 1.0 | 1.0 − 0.8 = 0.2 |
✔ Check: 0.5 + 0.1 + 0.2 + 0.2 = 1
Step 3: Formula for Mean
The mean (expected value) of a discrete random variable is:
E(X) = Σ x · P(X = x)
Step 4: Calculation of Mean
E(X) = (0)(0.5) + (1)(0.1) + (2)(0.2) + (3)(0.2)
E(X) = 0 + 0.1 + 0.4 + 0.6
E(X) = 1.1
Final Answer
The mean value of X is 1.1
Explanation of Possible Options
Option A: 1.0 ❌
This value ignores the contribution of X = 3, leading to an underestimated mean.
Option B: 1.1 ✅
Correct. This value is obtained by correctly converting the CDF into PMF and applying
the expectation formula.
Option C: 1.5 ❌
This assumes uniform probabilities, which is not supported by the given CDF.
Option D: 2.0 ❌
This incorrectly assigns higher probability to larger values of X.
Important Exam Tip
Always convert the CDF into a PMF before calculating the mean or variance.
Conclusion
The mean of the discrete random variable from cumulative probability
is 1.1. Careful extraction of the probability mass function from the
CDF is essential for solving such problems accurately in competitive examinations.
