Glossary

7.2 Functions of multiple random variables

3 min read•july 19, 2024

Functions of multiple random variables extend single-variable concepts to more complex scenarios. They involve transforming sets of random variables into new ones, using joint probability distributions and the Jacobian matrix to derive new distributions.

These functions are crucial in engineering, allowing us to model and analyze systems with multiple inputs. We can calculate expected values and variances for these functions, enabling us to solve real-world problems in communication systems, reliability analysis, and parameter estimation.

Functions of Multiple Random Variables

Functions of multiple random variables

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  • Consider nn random variables X1,X2,…,XnX_1, X_2, \ldots, X_n with a joint PDF fX1,X2,…,Xn(x1,x2,…,xn)f_{X_1, X_2, \ldots, X_n}(x_1, x_2, \ldots, x_n)
  • Define mm functions of these random variables as Y1=g1(X1,X2,…,Xn)Y_1 = g_1(X_1, X_2, \ldots, X_n), Y2=g2(X1,X2,…,Xn)Y_2 = g_2(X_1, X_2, \ldots, X_n), ..., Ym=gm(X1,X2,…,Xn)Y_m = g_m(X_1, X_2, \ldots, X_n)
  • Derive the joint PDF of Y1,Y2,…,YmY_1, Y_2, \ldots, Y_m using the joint PDF of X1,X2,…,XnX_1, X_2, \ldots, X_n and the Jacobian matrix
  • Extend the concepts of functions of one random variable to multiple random variables (sum, product, ratio)

Joint CDF and PDF determination

  • Calculate the joint CDF of Y1,Y2,…,YmY_1, Y_2, \ldots, Y_m given by FY1,Y2,…,Ym(y1,y2,…,ym)=P(Y1≤y1,Y2≤y2,…,Ym≤ym)F_{Y_1, Y_2, \ldots, Y_m}(y_1, y_2, \ldots, y_m) = P(Y_1 \leq y_1, Y_2 \leq y_2, \ldots, Y_m \leq y_m)
    • Integrate the joint PDF of X1,X2,…,XnX_1, X_2, \ldots, X_n over the region where g1(x1,x2,…,xn)≤y1g_1(x_1, x_2, \ldots, x_n) \leq y_1, g2(x1,x2,…,xn)≤y2g_2(x_1, x_2, \ldots, x_n) \leq y_2, ..., gm(x1,x2,…,xn)≤ymg_m(x_1, x_2, \ldots, x_n) \leq y_m
  • Obtain the joint PDF of Y1,Y2,…,YmY_1, Y_2, \ldots, Y_m by differentiating the joint CDF with respect to y1,y2,…,ymy_1, y_2, \ldots, y_m
  • Alternatively, derive the joint PDF using the joint PDF of X1,X2,…,XnX_1, X_2, \ldots, X_n and the Jacobian matrix JJ
    • fY1,Y2,…,Ym(y1,y2,…,ym)=fX1,X2,…,Xn(x1,x2,…,xn)⋅∣J∣f_{Y_1, Y_2, \ldots, Y_m}(y_1, y_2, \ldots, y_m) = f_{X_1, X_2, \ldots, X_n}(x_1, x_2, \ldots, x_n) \cdot |J|, where ∣J∣|J| is the absolute value of the determinant of the Jacobian matrix

Expected value and variance calculation

  • Calculate the g(X1,X2,…,Xn)g(X_1, X_2, \ldots, X_n) using the formula E[g(X1,X2,…,Xn)]=∫−∞∞⋯∫−∞∞g(x1,x2,…,xn)⋅fX1,X2,…,Xn(x1,x2,…,xn) dx1 dx2 … dxnE[g(X_1, X_2, \ldots, X_n)] = \int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} g(x_1, x_2, \ldots, x_n) \cdot f_{X_1, X_2, \ldots, X_n}(x_1, x_2, \ldots, x_n) \, dx_1 \, dx_2 \, \ldots \, dx_n
  • Determine the variance of a function g(X1,X2,…,Xn)g(X_1, X_2, \ldots, X_n) using the formula Var[g(X1,X2,…,Xn)]=E[g2(X1,X2,…,Xn)]−(E[g(X1,X2,…,Xn)])2Var[g(X_1, X_2, \ldots, X_n)] = E[g^2(X_1, X_2, \ldots, X_n)] - (E[g(X_1, X_2, \ldots, X_n)])^2
  • Apply the linearity of expectation to simplify calculations for sums of random variables
  • Use the properties of variance to simplify calculations for sums and products of independent random variables

Applications in engineering problems

  • Apply the concepts of functions of multiple random variables to solve engineering problems
    • Signal-to-noise ratio (SNR) in communication systems
      • Model signal power and noise power as random variables
      • SNR is a function of these random variables
    • Reliability analysis of systems with multiple components
      • Model component reliabilities as random variables
      • System reliability depends on these random variables
    • Estimation of parameters in statistical models
      • Estimated parameters are functions of the observed data (random variables)
  • Follow these steps to solve engineering problems involving functions of multiple random variables
    1. Identify the relevant random variables and their joint PDF
    2. Define the function(s) of interest in terms of the random variables
    3. Determine the joint CDF or joint PDF of the function(s) using the techniques described earlier
    4. Calculate the expected value, variance, or other relevant properties of the function(s)
    5. Interpret the results in the context of the engineering problem

Key Terms to Review (15)

Bivariate Distribution: A bivariate distribution is a probability distribution that describes the behavior of two random variables simultaneously. It provides a way to understand the relationship and dependencies between the two variables by detailing how their probabilities interact. This concept is crucial for exploring functions of multiple random variables, as it allows for the modeling and analysis of complex systems where outcomes depend on more than one variable.
Central Limit Theorem for Sums: The Central Limit Theorem for sums states that when you take the sum of a large number of independent and identically distributed random variables, the distribution of that sum will approach a normal distribution, regardless of the original distribution of the variables. This powerful theorem allows for simplifications in calculations and predictions when dealing with random variables, especially when analyzing their combined effects in various situations.
Change of Variables Technique: The change of variables technique is a method used in probability and statistics to simplify the analysis of functions of random variables by transforming them into new variables. This approach allows for easier calculation of probabilities, expected values, and distributions when dealing with multiple random variables, making it a valuable tool in understanding complex relationships between these variables.
Correlation Coefficient: The correlation coefficient is a statistical measure that quantifies the strength and direction of a linear relationship between two random variables. It ranges from -1 to 1, where values closer to 1 indicate a strong positive correlation, values closer to -1 indicate a strong negative correlation, and values around 0 suggest no linear correlation. This concept is vital for understanding relationships in various contexts, including random variables and their independence, joint distributions, and the analysis of functions involving multiple variables.
Covariance: Covariance is a measure of how much two random variables change together. It indicates the direction of the linear relationship between the variables, where a positive covariance means that as one variable increases, the other tends to increase as well, while a negative covariance indicates that as one variable increases, the other tends to decrease. This concept is essential in understanding joint distributions and functions of multiple variables, as it helps quantify their interdependence and is crucial for calculating expectations and variances.
Expected Value of a Function: The expected value of a function is a statistical measure that provides the average outcome of a random variable transformed by a specific function. It combines the probabilities of all possible outcomes with their respective values, allowing for the assessment of more complex scenarios involving multiple random variables. This concept plays a crucial role in decision-making processes where uncertainty is present, as it helps quantify potential gains or losses based on varying conditions.
Function of Random Variables: A function of random variables is a new random variable created by applying a mathematical function to one or more existing random variables. This concept allows for the manipulation and analysis of random variables in various applications, such as risk assessment and decision-making. By understanding how functions interact with these random variables, we can derive new probabilities, expectations, and variances that are essential in predicting outcomes in uncertain environments.
Independence of Random Variables: Independence of random variables occurs when the occurrence of one random variable does not affect the probability distribution of another. This concept is essential when working with functions of multiple random variables, as it simplifies calculations and allows for the use of product distributions when determining joint probabilities. Understanding independence helps in assessing the overall behavior of multiple variables and is crucial for applications like risk assessment and statistical inference.
Joint Probability Distribution: A joint probability distribution is a mathematical function that describes the likelihood of two or more random variables occurring simultaneously. It provides a comprehensive view of how the variables interact, allowing for the calculation of probabilities associated with specific outcomes for each variable. This concept is crucial for understanding relationships between multiple random variables and is foundational for deriving marginal and conditional distributions.
Law of Total Expectation: The Law of Total Expectation states that the expected value of a random variable can be found by taking the weighted average of its conditional expected values given different scenarios. This principle connects various concepts, allowing one to break down complex expectations into simpler, more manageable parts by conditioning on different events or random variables.
Linear Transformation: A linear transformation is a mathematical operation that maps a vector space into another vector space while preserving the operations of vector addition and scalar multiplication. This concept is crucial in understanding how random variables can change and interact, as it allows for the analysis of multiple variables through transformations that maintain linear relationships. In contexts involving one or multiple random variables, linear transformations can simplify complex relationships and facilitate the computation of probabilities.
Marginal Probability: Marginal probability refers to the probability of a single event occurring without regard to any other events. It is calculated by summing or integrating the joint probabilities of the event with respect to the other variable(s). This concept is crucial for understanding how individual events behave in relation to a larger set of outcomes and is linked to various important principles in probability, such as total probability, independence of variables, and functions involving multiple variables.
Monte Carlo Simulation: Monte Carlo Simulation is a computational technique that uses random sampling to estimate mathematical functions and simulate the behavior of complex systems. By generating a large number of random samples, it helps in understanding the impact of risk and uncertainty in various scenarios, including those involving multiple random variables, different probability distributions, and stochastic processes.
Multivariate normal distribution: A multivariate normal distribution is a probability distribution that describes multiple random variables that are normally distributed and possibly correlated with each other. It is characterized by a mean vector and a covariance matrix, which together define the shape and orientation of the distribution in multidimensional space. This distribution plays a crucial role in statistical modeling, especially in cases where multiple variables interact with each other.
Variance-Covariance Matrix: The variance-covariance matrix is a square matrix that provides a summary of the variances and covariances among multiple random variables. This matrix helps to understand the relationships between these variables, showing how much they change together and how much variability there is in each variable. It's crucial when working with functions of multiple random variables as it helps to describe their joint behavior and is foundational for concepts like multivariate normal distributions.
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