2.4 Expected value and variance
2 min read•july 25, 2024
Probability and statistics form the backbone of information theory. These tools help us understand random events and measure uncertainty in data transmission and compression.
, linearity, , and common distributions are key concepts. They allow us to analyze communication systems, estimate code lengths, and model noise in channels. These fundamentals are crucial for optimizing information processing and transmission.
Probability and Statistics Fundamentals
Expected value calculation
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- Expected value (E[X]) measures central tendency for random variable represents average outcome if experiment repeated many times
- Discrete random variables:
- Continuous random variables:
- Properties: E[c] = c (c is constant), E[X + Y] = E[X] + E[Y], E[cX] = c · E[X] (c is constant)
- Applications estimate average code length in data compression and analyze performance of communication systems
Linearity of expectation
- Linearity property holds true even when X and Y are dependent
- Simplifies complex calculations and analyzes systems with multiple components
- Examples include expected value of sum of dice rolls and average number of successful transmissions in communication channel
- Relates to :
- Extends to multiple variables: E[X1 + X2 + ... + Xn] = E[X1] + E[X2] + ... + E[Xn]
Variance and standard deviation
- Variance (Var(X) or σ²) measures dispersion or spread of random variable
- Calculation: or
- is square root of variance, same units as random variable
- Properties: Var(c) = 0 (c is constant), Var(aX + b) = a²Var(X) (a and b are constants)
- provides bounds on probability based on variance
- Applications quantify uncertainty in data transmission and analyze error rates in communication systems
Expectations for common distributions
- Bernoulli: E[X] = p, Var(X) = p(1-p)
- Binomial: E[X] = np, Var(X) = np(1-p)
- Poisson: E[X] = λ, Var(X) = λ
- Uniform (continuous): E[X] = (a + b) / 2, Var(X) = (b - a)² / 12
- Exponential: E[X] = 1 / λ, Var(X) = 1 / λ²
- Normal: E[X] = μ, Var(X) = σ²
- Applications model noise in communication channels, analyze arrival times of data packets, estimate probabilities of rare events in error correction
Key Terms to Review (17)
Bernoulli distribution: The Bernoulli distribution is a discrete probability distribution that models a random experiment with only two possible outcomes: success (1) or failure (0). This distribution is fundamental in probability theory and serves as the basis for binomial distributions, providing insights into events with binary outcomes.
Binomial Distribution: A binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by two parameters: the number of trials, denoted as 'n', and the probability of success on each trial, denoted as 'p'. Understanding this distribution helps in calculating probabilities related to discrete outcomes and analyzing events with binary results.
Chebyshev's Inequality: Chebyshev's Inequality is a statistical theorem that provides a bound on the probability that a random variable deviates from its mean. This inequality states that for any real number $k > 1$, at least $1 - \frac{1}{k^2}$ of the values of a dataset lie within $k$ standard deviations of the mean. It connects expected value and variance by emphasizing how spread out values can be in relation to these statistical measures.
Continuous Random Variable: A continuous random variable is a type of random variable that can take on an infinite number of possible values within a given range or interval. This contrasts with discrete random variables, which can only assume distinct, separate values. Continuous random variables are essential in probability theory as they require specific mathematical tools, such as probability density functions, to describe their behavior and calculate probabilities.
Covariance: Covariance is a statistical measure that indicates the extent to which two random variables change together. It helps in understanding the relationship between the variables, whether they tend to increase or decrease in tandem. The concept is crucial for assessing the degree of correlation, and it plays a significant role in calculating variance when dealing with multiple random variables.
Discrete Random Variable: A discrete random variable is a type of variable that can take on a countable number of distinct values, often representing counts or categories. These variables are used to model scenarios where outcomes are specific and separate, such as the number of heads in a series of coin flips or the number of students in a classroom. Discrete random variables are fundamental to understanding how probabilities are assigned and calculated in various scenarios.
E(x) = ∫ [x * f(x) dx]: The expression e(x) = ∫ [x * f(x) dx] represents the expected value of a continuous random variable, where x is the variable and f(x) is its probability density function (PDF). This concept is crucial as it quantifies the average outcome one can expect from a random process, providing insights into its behavior and properties. Understanding expected value helps in assessing risk, making predictions, and drawing conclusions from data in various fields.
E(x) = σ [x * p(x)]: The equation e(x) = σ [x * p(x)] represents the expected value of a random variable x, where p(x) is the probability of x occurring. The expected value is a crucial concept that provides a measure of the central tendency of a random variable, indicating where the values are most likely to cluster. This term is fundamentally connected to variance, which measures how much the values deviate from the expected value.
E[ax + by] = ae[x] + be[y]: The equation e[ax + by] = ae[x] + be[y] expresses a fundamental property of the expected value in probability theory, showcasing how the expected value of a linear combination of random variables can be calculated. This property demonstrates that expectation is a linear operator, which means you can separate the expected values of the individual components when they are scaled by constants. Understanding this concept is crucial for analyzing distributions and making predictions based on random variables.
E[xy] = e[x]e[y] + cov(x,y): This equation describes the relationship between the expected value of the product of two random variables and their individual expected values along with their covariance. It shows how the expected value of a joint distribution can be expressed in terms of individual distributions and their correlation. Understanding this relationship is crucial when analyzing dependencies between variables and calculating variances.
Expected value: Expected value is a fundamental concept in probability that represents the average outcome of a random variable over many trials. It connects the likelihood of different outcomes with their associated values, helping to summarize a probability distribution. Understanding expected value is crucial for analyzing decision-making processes and assessing risk, as it provides a single number that encapsulates what one can anticipate from a probabilistic scenario.
Linearity of Expectation: Linearity of expectation is a fundamental property in probability theory that states the expected value of the sum of random variables is equal to the sum of their expected values, regardless of whether the random variables are independent or not. This concept simplifies calculations in various scenarios by allowing for straightforward manipulation of expected values, which is particularly useful when dealing with complex problems involving multiple variables.
Normal Distribution: Normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. This bell-shaped curve is significant in statistics because it describes how many types of real-valued random variables tend to be distributed, and it plays a crucial role in statistical inference and hypothesis testing.
Poisson distribution: The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, assuming these events happen with a known constant mean rate and independently of the time since the last event. This distribution is particularly useful in modeling random events that occur independently, like phone calls received at a call center or decay events from a radioactive source.
Standard Deviation (σ): Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of data values. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation signifies that the data points are spread out over a wider range of values. Understanding standard deviation is crucial in assessing the reliability of the expected value and variance, as it provides insights into the consistency and predictability of a random variable's outcomes.
Uniform Distribution: A uniform distribution is a probability distribution where all outcomes are equally likely within a specified range. This means that every event in the set has the same chance of occurring, leading to a flat probability function. Uniform distributions can be discrete, where a finite number of outcomes exist, or continuous, where the possible outcomes form an interval on the real line. This concept plays a crucial role in various areas such as statistics, information theory, and data analysis.
Variance: Variance, represented by the formula $$var(x) = e[x²] - (e[x])²$$, quantifies how much a random variable differs from its expected value. It provides a measure of the spread or dispersion of a set of values, indicating how far the individual values are from the mean. Understanding variance is crucial in statistics as it helps in assessing the reliability and variability of data.