Finding the Probability that AB ≤ 1/4 Using the Gamma Distribution

This article explores the problem of determining the probability that the product of two uniformly distributed random variables AB ≤ 1/4. We will employ integrating techniques within the unit square [0,1] × [0,1] and leverage the Gamma distribution to transform the problem into a more straightforward integral. Understanding how to solve such problems can be crucial in various fields including statistics and data science.

1. Introduction to the Problem

Random variables A and B are chosen uniformly from the interval [0,1]. We aim to find the probability that their product AB is less than or equal to 1/4. This involves visualizing the unit square [0,1] × [0,1] and working with the inequality AB ≤ 1/4.

2. Visualizing the Problem

The unit square [0,1] × [0,1] is the domain for A and B. To find the area where AB ≤ 1/4, we can rewrite the inequality as B ≤ 1/(4A). This inequality defines a curve in the unit square. Let's calculate the area under this curve.

3. Calculating the Area

We will split the calculation into two regions:

Region 1: When A ≤ 1/4, B can take any value from 0 to 1. Region 2: When A > 1/4, B is defined by the curve B 1/(4A).

3.1 Region 1

For A in the range [0, 1/4], B can take any value from 0 to 1. The area of this region is:

[ text{Area}_1 int_0^{1/4} 1 , dA frac{1}{4} ]

3.2 Region 2

For A in the range (1/4, 1), the area under the curve B 1/(4A) is given by:

[ text{Area}_2 int_{1/4}^{1} frac{1}{4A} , dA ]

We compute this integral as follows:

[ text{Area}_2 frac{1}{4} int_{1/4}^{1} frac{1}{A} , dA frac{1}{4} left[ ln A right]_{1/4}^{1} ]

Evaluating this, we get:

[ text{Area}_2 frac{1}{4} left( ln 1 - ln frac{1}{4} right) frac{1}{4} left( 0 - ln frac{1}{4} right) frac{1}{4} ln 4 ]

Simplifying further:

[ ln 4 2 ln 2 ]

Hence:

[ text{Area}_2 frac{1}{4} cdot 2 ln 2 frac{1}{2} ln 2 ]

3.3 Total Area

The total area where AB ≤ 1/4 is the sum of the two areas:

[ text{Total Area} text{Area}_1 text{Area}_2 frac{1}{4} frac{1}{2} ln 2 ]

3.4 Calculating the Probability

The probability that AB ≤ 1/4 is the total area divided by the area of the unit square, which is 1:

[ P(AB leq frac{1}{4}) frac{text{Total Area}}{1} frac{1}{4} frac{1}{2} ln 2 ]

Expressed as:

[ P(AB leq frac{1}{4}) frac{1}{4} frac{1}{2} ln 2 ]

4. Proof Using the Gamma Distribution

The Gamma distribution plays a significant role in transforming the problem. We can model -LnA and -LnB as Gamma distributions:

4.1 Proof of 1: Transforming A and B

Let Y -LnA, where Y ~ Gamma(1,1). The cumulative distribution function (CDF) is:

[ F_Y(y) P(Y leq y) P(-ln A leq y) P(A leq e^{-y}) ]

For A ~ U[0,1], we have:

[ F_Y(y) e^{-y} text{ for } y in (-infty, 0] ]

The probability density function (PDF) of Y is:

[ f_Y(y) e^{-y} text{ for } y in [0, infty) ]

;p

4.2 Proof of 2: Joint Distribution of -LnA and -LnB

Let X_1 -LnA and X_2 -LnB, which are independent Gamma(1,1) distributions. The joint distribution of X_1 and X_2 is given by:

[ f_{X_1, X_2}(x_1, x_2) e^{-x_1 - x_2} text{ for } x_1, x_2 geq 0 ]

For Z X_1 X_2, the probability density function of Z is:

[ f_Z(z) z e^{-z} text{ for } z geq 0 ]

Thus, the probability that -LnA - LnB X_1 - X_2 ≥ -ln4 is:

[ P(X_1 - X_2 geq -ln4) int_{-ln4}^{infty} z e^{-z} dz frac{1}{4}[ln4 - 1] approx 0.597 ]

5. Conclusion

The problem of determining the probability that AB ≤ 1/4 can be effectively solved by visualizing the unit square, splitting it into regions, and integrating. Additionally, using the properties of the Gamma distribution simplifies the calculations. Familiarizing oneself with these distributions and their properties can greatly enhance the ability to tackle complex statistical and probabilistic problems.