Calculating the Variance of a Random Variable Using Cumulative Distribution Function

In probability and statistics, the variance of a random variable is an important measure of variability. It quantifies how much the values of the random variable deviate from the mean. While there are various methods to compute variance, this article provides a detailed approach to finding the variance of a random variable using its cumulative distribution function (CDF).

Steps to Calculate Variance Using CDF

To find the variance of a random variable X from its cumulative distribution function (CDF) F(x), follow these steps:

Identify the Probability Density Function (PDF)

Depending on whether the random variable X is continuous or discrete:

Continuous Variables: Differentiate the CDF to obtain the probability density function (PDF) f(x). Discrete Variables: Identify the probabilities from the CDF.

Calculate the Expected Value (Mean) E(X)

The expected value (mean) of X can be calculated using the following formulas for continuous and discrete variables:

Continuous Variables:

E(X) ∫∞-∞ x f(x) dx

Discrete Variables:

E(X) Σ xi P(X xi)

Calculate the Expected Value of X2

The expected value of X2 can be calculated using similar formulas, again for both continuous and discrete variables:

Continuous Variables:

E(X2) ∫∞-∞ x2 f(x) dx

Discrete Variables:

E(X2) Σ xi2 P(X xi)

Calculate the Variance

The variance of X is given by:

Var(X) E(X2) - E(X)2

Special Case: Non-negative Random Variables

When the random variable X is non-negative (e.g., in an exponential distribution), there is a neat trick to compute the variance using the CDF function F(x):

Var(X) E(X2) - E(X)2 ∫∞0 x2 (1 - F(x)) dx - ∫∞0 (1 - F(x)) dx2

This formula can be derived using integration by parts for the expected value of X2 and X.

Integration by Parts Derivation

For the expected value of X2:

Let u x2 and dv f(x) dx v F(x) - 1 du 2x dx

The integral becomes:

E(X2) x2 F(x) - 1 |∞0 - ∫∞0 2x (1 - F(x)) dx

This simplifies to:

E(X2) ∫∞0 2x (1 - F(x)) dx

Similarly, for the expected value of X:

E(X) ∫∞0 (1 - F(x)) dx

This is equivalent to the continuous version of the sum of tail probabilities for discrete random variables.

Practical Considerations

These formulas are particularly useful when dealing with non-negative random variables. For instance, in the exponential distribution, the variance can be easily computed using the CDF, simplifying the calculations significantly.

Conclusion

Understanding the relationship between the cumulative distribution function and variance is crucial for statisticians, data scientists, and researchers. By leveraging these formulas, one can efficiently calculate variance without needing to fully derive the probability density function (PDF).

References

Probability and Statistics for Engineers and Scientists (9th Edition) by Walpole, Myers, Myers, and Ye. Introduction to Probability by Dimitri P. Bertsekas and John N. Tsitsiklis.

Keywords

cumulative distribution function, variance, probability density function