🎲Intro to Probabilistic Methods Unit 6 – Random Variable Functions

Key Concepts and Definitions

• Random variable is a function that assigns a numerical value to each outcome in a sample space
• Probability distribution describes the likelihood of different values occurring for a random variable
• Discrete random variables have countable values (integers) while continuous random variables have uncountable values (real numbers)
• Cumulative distribution function (CDF) gives the probability that a random variable is less than or equal to a specific value
• Probability density function (PDF) for continuous random variables represents the relative likelihood of the variable taking on a given value
• Probability mass function (PMF) for discrete random variables gives the probability of the variable being equal to a specific value
• Expected value is the average value of a random variable over many trials, calculated as the sum of each value multiplied by its probability
• Variance measures the spread or dispersion of a random variable around its expected value, calculated as the average squared deviation from the mean

Types of Random Variables

• Bernoulli random variable has only two possible outcomes, typically labeled as success (1) and failure (0)
• Binomial random variable counts the number of successes in a fixed number of independent Bernoulli trials (coin flips)
• Poisson random variable models the number of events occurring in a fixed interval of time or space (customer arrivals)
• Geometric random variable represents the number of trials needed to achieve the first success in a series of independent Bernoulli trials
• Uniform random variable has equal probability of taking on any value within a specified range (rolling a fair die)
• Normal (Gaussian) random variable is characterized by a bell-shaped curve and is determined by its mean and standard deviation
• Exponential random variable models the time between events in a Poisson process (waiting times)

Probability Distributions

• Probability distributions assign probabilities to the possible values of a random variable
• Discrete probability distributions include Bernoulli, binomial, Poisson, and geometric
  • Bernoulli distribution has probability $p$ for success and $1-p$ for failure
  • Binomial distribution has parameters $n$ (number of trials) and $p$ (probability of success) with PMF $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$
  • Poisson distribution has parameter $\lambda$ (average number of events) with PMF $P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}$
• Continuous probability distributions include uniform, normal, and exponential
  • Uniform distribution has equal probability density over a specified range $[a,b]$ with PDF $f(x) = \frac{1}{b-a}$ for $a \leq x \leq b$
  • Normal distribution has parameters $\mu$ (mean) and $\sigma$ (standard deviation) with PDF $f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$
  • Exponential distribution has parameter $\lambda$ (rate) with PDF $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$
• Joint probability distributions describe the probabilities of two or more random variables occurring together
• Marginal probability distributions are obtained by summing or integrating the joint distribution over the other variables

Functions of Random Variables

• Functions of random variables transform the values of a random variable into a new random variable
• If $X$ is a discrete random variable and $Y=g(X)$, then the PMF of $Y$ is given by $P(Y=y) = \sum_{x:g(x)=y} P(X=x)$
• If $X$ is a continuous random variable and $Y=g(X)$, then the PDF of $Y$ is given by $f_Y(y) = f_X(g^{-1}(y)) \cdot \left|\frac{d}{dy}g^{-1}(y)\right|$
• Linear functions of random variables $(Y=aX+b)$ preserve the type of distribution with modified parameters
  • For a normal random variable $X$, if $Y=aX+b$, then $Y$ is also normally distributed with $\mu_Y=a\mu_X+b$ and $\sigma_Y=|a|\sigma_X$
• Functions of multiple random variables combine the values of several random variables into a new random variable
  • If $X$ and $Y$ are independent random variables, then the expected value of their sum is the sum of their expected values: $E[X+Y] = E[X] + E[Y]$
  • If $X$ and $Y$ are independent random variables, then the variance of their sum is the sum of their variances: $Var(X+Y) = Var(X) + Var(Y)$

Expected Value and Variance

• Expected value (mean) of a discrete random variable $X$ is calculated as $E[X] = \sum_{x} x \cdot P(X=x)$
• Expected value of a continuous random variable $X$ is calculated as $E[X] = \int_{-\infty}^{\infty} x \cdot f_X(x) dx$
• Variance of a discrete random variable $X$ is calculated as $Var(X) = E[(X-E[X])^2] = \sum_{x} (x-E[X])^2 \cdot P(X=x)$
• Variance of a continuous random variable $X$ is calculated as $Var(X) = E[(X-E[X])^2] = \int_{-\infty}^{\infty} (x-E[X])^2 \cdot f_X(x) dx$
• Standard deviation is the square root of the variance and measures the spread of the distribution
• Properties of expected value include linearity: $E[aX+b] = aE[X] + b$ for constants $a$ and $b$
• Properties of variance include $Var(aX+b) = a^2Var(X)$ for constants $a$ and $b$

Transformations of Random Variables

• Linear transformations of the form $Y=aX+b$ change the location and scale of the distribution
  • For a normal random variable $X$, if $Y=aX+b$, then $Y$ is also normally distributed with $\mu_Y=a\mu_X+b$ and $\sigma_Y=|a|\sigma_X$
  • For an exponential random variable $X$ with parameter $\lambda$, if $Y=aX$, then $Y$ is also exponentially distributed with parameter $\lambda/a$
• Nonlinear transformations can change the type of distribution
  • If $X$ is a standard normal random variable, then $Y=X^2$ follows a chi-squared distribution with 1 degree of freedom
  • If $X$ is a uniform random variable on $[0,1]$, then $Y=-\ln(X)/\lambda$ follows an exponential distribution with parameter $\lambda$
• Convolution is used to find the distribution of the sum of independent random variables
  • If $X$ and $Y$ are independent continuous random variables with PDFs $f_X(x)$ and $f_Y(y)$, then the PDF of $Z=X+Y$ is given by the convolution integral $f_Z(z) = \int_{-\infty}^{\infty} f_X(x)f_Y(z-x) dx$

Applications and Examples

• Modeling stock prices using normal random variables to capture the logarithmic returns
• Analyzing the number of defective items in a production line using a binomial distribution
• Predicting the waiting time between customer arrivals at a service center using an exponential distribution
• Assessing the probability of a component failing within a given time frame using a Weibull distribution
• Estimating the time taken to complete a project using a gamma distribution to model the sum of independent exponential tasks
• Evaluating the reliability of a system with multiple components using functions of random variables
• Determining the optimal inventory level by minimizing the expected total cost, considering demand as a random variable

Common Pitfalls and Tips

• Ensure the random variable is clearly defined and the sample space is properly identified
• Be cautious when assuming independence between random variables, as it may not always hold true
• Remember to use the appropriate probability distribution for the given problem context
• When transforming random variables, ensure that the function is bijective (one-to-one and onto) to avoid complications
• Double-check the limits of integration or summation when calculating expected values and variances
• Pay attention to the domain of the function when transforming random variables to avoid undefined values
• When working with joint distributions, ensure that the correct variables are being marginalized or conditioned on
• Verify that the necessary conditions for applying specific theorems or properties are met before using them