Independence and Conditioning#

1. Theory#

In this part we will discuss two topics:

1. Independence
2. Conditional probability

Independence is a fundamental concept that describes the relationship between two or more random variables. When two random variables, X and Y, are independent, it means that the occurrence or value of one variable does not influence the occurrence or value of the other variable.

Formally, X and Y are considered independent if and only if the joint probability function (or cumulative distribution function) can be factorized into the product of their marginal probability functions (or cumulative distribution functions).

Mathematically, for independent variables X and Y, the following relationship holds:

\(F(x, y) = P(x<X \bigcap y<Y ) = P(x<X)P(y<Y) = F(x)F(y)\)

The different relationships above highlights the connection between the joint cumulative distribution function (CDF) and the marginal CDFs of two independent random variables, X and Y.

Equations explanation!

  1. \(F(x, y) = P(x<X \bigcap y<Y)\): This equation represents the joint cumulative distribution function, which gives us the probability that both X and Y are less than their respective values, x and y. It denotes the cumulative probability of the joint event where X is less than x and Y is less than y simultaneously.

  2. \(F(x, y) = P(x<X)P(y<Y)\): This equation expresses the joint cumulative distribution function in terms of the marginal probabilities of X and Y. It states that the joint probability of X being less than x and Y being less than y can be obtained by multiplying the probabilities of X being less than x and Y being less than y individually. This holds true when X and Y are independent.

  3. \(F(x, y) = F(x)F(y)\): This equation further simplifies the relationship between the joint cumulative distribution function and the marginal CDFs. It indicates that the joint cumulative distribution function of X and Y can be factorized into the product of the marginal cumulative distribution functions of X and Y. This factorization is valid when X and Y are independent.

Conditional probability is a measure of the probability of an event occurring given that another event has already occurred. The following relationship holds for events X and Y:

\(P(x<X \vert y<Y) = \frac {P(x<X \bigcap y<Y )}{P(y<Y)} = \frac{F(x, y)}{F(y)}\)

\( F(x, y) = F(x \vert y)F(y) = F(y \vert x)F(x) \)

Where:

  1. \(P(x<X \vert y<Y)\) represents the conditional probability that event X is less than a certain value x, given that event Y is less than a certain value y.

  2. \(P(x<X \bigcap y<Y)\) represents the joint probability that both X is less than x and Y is less than y.

  3. \(F(x, y)\) represents the joint cumulative distribution function (CDF) of X and Y, which gives the probability that both X and Y are less than or equal to their respective values x and y.

  4. \(F(x)\) represents the marginal cumulative distribution function (CDF) of X, which gives the probability that X is less than or equal to x.

  5. \(F(y)\) represents the marginal cumulative distribution function (CDF) of Y, which gives the probability that Y is less than or equal to y.

Equations explanation!

  1. \(P(x<X \vert y<Y) = \frac {P(x<X \bigcap y<Y )}{P(y<Y)}\) This formula represents the definition of conditional probability. The probability of X being less than x, given that Y is less than y, is equal to the joint probability of X being less than x and Y being less than y, divided by the probability of Y being less than y.

  2. \(F(x, y) = F(x \vert y)F(y)\) This formula expresses the joint CDF of X and Y in terms of the conditional CDF of X given Y, multiplied by the marginal CDF of Y. It states that the joint probability of X being less than or equal to x and Y being less than or equal to y is equal to the probability of X being less than or equal to x, given that Y is less than or equal to y, multiplied by the probability of Y being less than or equal to y.

  3. \(F(x, y) = F(y \vert x)F(x)\) This formula expresses the joint CDF of X and Y in terms of the conditional CDF of Y given X, multiplied by the marginal CDF of X. It states that the joint probability of X being less than or equal to x and Y being less than or equal to y is equal to the probability of Y being less than or equal to y, given that X is less than or equal to x, multiplied by the probability of X being less than or equal to x.

2. Probability Analysis in Civil Engineering#

Scenario 1: Independent X and Y

In a civil engineering project, engineers investigate the properties of two measured quantities, X and Y. Theoretical cumulative distribution functions, F(x) and G(y), have been fitted to the data obtained from years of measurements. Assuming X and Y are independent, the engineers can perform probability analysis to gain insights into structural integrity.

The cumulative distribution function for X is given by:

\[\begin{split}F(x) = \left\{ \begin{array}{ll} 0 & x < -1 \\ \frac{x+1}{2} & -1 \leq x \leq 1 \\ 1 & x > 1 \end{array} \right.\end{split}\]

The cumulative distribution function for Y is given by:

\[\begin{split}G(y) = \left\{ \begin{array}{ll} 0 & y < 0 \\ 1 - e^{-y} & y \geq 0 \end{array} \right.\end{split}\]

Let’s explore some probability scenarios.

Scenario 2: Probability of X and Y Exceeding Thresholds

1. What is the probability that X is less than or equal to -0.1 and Y is less than or equal to 1?

Since X and Y are independent, we can calculate this probability as:

\[P(X \leq -0.1, Y \leq 1) = F(-0.1) \cdot G(1) = \frac{-0.1 + 1}{2} \cdot (1 - e^{-1}) = 0.28\]

2. What is the probability that X is greater than 0.8 given that Y is greater than 10?

Since X and Y are independent, the probability can be calculated as:

\[P(X > 0.8 | Y > 10) = P(X > 0.8) = 1 - F(0.8) = 1 - \frac{0.8 + 1}{2} = 0.1\]

Scenario 3: Accounting for Joint Distribution

Further analysis reveals the joint distribution H(x, y) that accurately represents the interdependence between X and Y.

3. What is the probability that X is greater than or equal to 0.8 and Y is greater than or equal to 4.6?

The joint distribution allows us to evaluate this probability as:

\[P(X \geq 0.8, Y \geq 4.6) = 1 - (F(0.8) + G(4.6) - H(0.8, 4.6))\]

By substituting the values, we find:

\[P(X \geq 0.8, Y \geq 4.6) = 1 - \left(\frac{0.8 + 1}{2} + (1 - e^{-4.6}) - \frac{(0.8 + 1)(e^{4.6} - 1)}{0.8 + 2e^{4.6} - 1}\right) = 0.0019\]

4. What is the probability that X is greater than or equal to 0.8 given that Y is greater than or equal to 4.6?

Using the joint distribution, we can calculate this probability as:

\[P(X \geq 0.8 | Y \geq 4.6) = \frac{P(X \geq 0.8, Y \geq 4.6)}{P(Y \geq 4.6)}\]

By substituting the values, we find:

\[P(X \geq 0.8 | Y \geq 4.6) = \frac{0.0019}{1 - G(4.6)} = 0.1931\]

These probability analyses enable civil engineers to make informed decisions about material behavior, load capacities, and structural reliability. By considering the independence or dependence between variables, they can design and construct robust and safe structures.

3.Probability Scenarios in Civil Engineering#

In civil engineering, understanding the probabilistic behavior of variables is crucial for making informed decisions and ensuring the safety and reliability of structures. Let’s explore some probability scenarios related to two variables, X and Y, which have significance in civil engineering applications. The joint cumulative distribution function (CDF) of X and Y is denoted as \(F_{XY}(x, y)\), and the marginal distribution functions of Y and X are denoted as \(F_Y(y)\) and \(F_X(x)\), respectively.

Consider the following functions:

\[F_{XY}(x, y) = \frac{1}{9}x^4y + \frac{1}{6}x^2y^2 + \frac{1}{3}x^3y^2 + \frac{1}{3}xy^3 + y^2\]
\[F_Y(y) = \frac{4}{9}x^3y + \frac{1}{3}xy^2 + x^2y^2 + \frac{1}{3}y^3\]
\[F_X(x) = \frac{1}{9}x^4 + \frac{1}{3}x^2y + \frac{2}{3}x^3y + xy^2 + 2y\]

Let’s explore some probability scenarios related to civil engineering using these functions:

1. Probability of X and Y within given ranges#

In civil engineering, it is often necessary to analyze the behavior of multiple variables within specified ranges. Suppose X represents the compressive strength of a concrete sample and Y represents the load applied to a structural member. What is the probability that the compressive strength X is less than or equal to 0.5 units (e.g. MPa) and the load Y is less than or equal to 0.7 units (e.g. kN)?

2. Conditional Probability#

In civil engineering, understanding conditional probabilities is crucial for assessing the performance of structures under specific conditions. Suppose X represents the deflection of a beam and Y represents the span length. What is the probability that X is greater than 0.3 units given that Y is equal to 0.6 units?

3. Joint Probability#

Another important aspect in civil engineering is evaluating the joint probability of variables X and Y. What is the probability that X, representing the soil settlement, is greater than or equal to 0.4 units (e.g. mm) and Y, representing the applied surcharge, is less than or equal to 2 units (e.g. kPa)?

4. Conditional Joint Probability#

Understanding conditional joint probabilities is essential in civil engineering for evaluating the behavior of multiple variables under specific conditions. What is the probability that X, representing the deflection of a structural element, is greater than or equal to 0.5 units given that Y, representing the applied load, is greater than or equal to 0.8 units?

Answers!

1. Probability of X and Y within given ranges:

To find the probability that X is less than or equal to 0.5 units (e.g. MPa) and Y is less than or equal to 0.7 units (e.g. kN), we need to evaluate the joint cumulative distribution function (CDF) \(F_{XY}(x, y)\) at these values:

\(P(X \leq 0.5, Y \leq 0.7) = F_{XY}(0.5, 0.7)\)

Substituting the values into the given CDF formula: \(P(X \leq 0.5, Y \leq 0.7) = \left(\frac{1}{9}\right)(0.5^4)(0.7) + \left(\frac{1}{6}\right)(0.5^2)(0.7^2) + \left(\frac{1}{3}\right)(0.5^3)(0.7^2) \) \(+ \left(\frac{1}{3}\right)(0.5)(0.7^3) + (0.7^2)\)

Calculating the expression: \(P(X \leq 0.5, Y \leq 0.7) = \left(\frac{0.5^4 \cdot 0.7}{9}\right) + \left(\frac{0.5^2 \cdot 0.7^2}{6}\right) + \left(\frac{0.5^3 \cdot 0.7^2}{3}\right) + \left(\frac{0.5 \cdot 0.7^3}{3}\right) + 0.7^2\)

Simplifying the calculations: \(P(X \leq 0.5, Y \leq 0.7) = 0.04167 + 0.02917 + 0.05972 + 0.03333 + 0.49\)

Calculating the final result: \(P(X \leq 0.5, Y \leq 0.7) = 0.65389\)

Therefore, the probability that X is less than or equal to 0.5 units and Y is less than or equal to 0.7 units is approximately 0.65389.

** 2.Conditional Probability:**

To calculate the probability that X is greater than 0.3 units given that Y is equal to 0.6 units, we need to consider the joint probability and the marginal probability of Y:

\(P(X > 0.3 | Y = 0.6) = \frac{P(X > 0.3, Y = 0.6)}{P(Y = 0.6)}\)

Assuming \(P(X > 0.3, Y = 0.6) = 0.1\) and \(P(Y = 0.6) = 0.4\), we can calculate the conditional probability.

\(P(X > 0.3 | Y = 0.6) = \frac{0.1}{0.4}\)

Simplifying the calculation: \(P(X > 0.3 | Y = 0.6) = 0.25\)

Therefore, the probability that X is greater than 0.3 units given that Y is equal to 0.6 units is 0.25.

3. Joint Probability:

The probability that X, representing the soil settlement, is greater than or equal to 0.4 units (e.g. mm), and Y, representing the applied surcharge, is less than or equal to 2 units (e.g. kPa), can be calculated by evaluating the joint cumulative distribution function (CDF) \(F_{XY}(x, y)\) at these values:

\(P(X \geq 0.4, Y \leq 2) = F_{XY}(0.4, 2)\)

Substituting the values into the given CDF formula: \(P(X \geq 0.4, Y \leq 2) = \left(\frac{1}{9}\right)(0.4^4)(2) + \left(\frac{1}{6}\right)(0.4^2)(2^2) + \left(\frac{1}{3}\right)(0.4^3)(2^2) + \left(\frac{1}{3}\right)(0.4)(2^3) + (2^2)\)

Calculating the expression: \(P(X \geq 0.4, Y \leq 2) = \left(\frac{0.4^4 \cdot 2}{9}\right) + \left(\frac{0.4^2 \cdot 2^2}{6}\right) + \left(\frac{0.4^3 \cdot 2^2}{3}\right) + \left(\frac{0.4 \cdot 2^3}{3}\right) + 2^2\)

Simplifying the calculations: \(P(X \geq 0.4, Y \leq 2) = 0.03352 + 0.10667 + 0.21333 + 0.21333 + 4\)

Calculating the final result: \(P(X \geq 0.4, Y \leq 2) = 4.56685\)

Therefore, the probability that X is greater than or equal to 0.4 units and Y is less than or equal to 2 units is approximately 4.56685.

4. Conditional Joint Probability:

The conditional joint probability that X, representing the deflection of a structural element, is greater than or equal to 0.5 units given that Y, representing the applied load, is greater than or equal to 0.8 units, can be calculated by considering the joint probability and the marginal probability of Y:

\(P(X \geq 0.5 | Y \geq 0.8) = \frac{P(X \geq 0.5, Y \geq 0.8)}{P(Y \geq 0.8)}\)

Assuming \(P(X \geq 0.5, Y \geq 0.8) = 0.2\) and \(P(Y \geq 0.8) = 0.6\), we can calculate the conditional joint probability.

\(P(X \geq 0.5 | Y \geq 0.8) = \frac{0.2}{0.6}\)

Simplifying the calculation: \(P(X \geq 0.5 | Y \geq 0.8) = 0.33333\)

Therefore, the conditional joint probability that X is greater than or equal to 0.5 units given that Y is greater than or equal to 0.8 units is approximately 0.33333.

Please note that the assumed values for the probabilities were arbitrary and provided for the purpose of demonstration. In practice, you would need to replace these values with the actual probabilities based on the specific problem or scenario you are working on.