Expected Value Formulas to Know for Intro to Probability

Expected value (EV) is a key concept in probability, representing the average outcome of random variables. It helps us understand distributions, whether discrete or continuous, and simplifies calculations through properties like linearity and conditional expectations.

1. Expected Value of a Discrete Random Variable

   • The expected value (EV) is the long-term average or mean of a random variable's possible values, weighted by their probabilities.
   • It is calculated using the formula: ( E(X) = \sum (x_i \cdot P(x_i)) ), where ( x_i ) are the values and ( P(x_i) ) are their probabilities.
   • The expected value provides a measure of the center of the distribution of the random variable.

2. Expected Value of a Continuous Random Variable

   • For continuous random variables, the expected value is determined using an integral: ( E(X) = \int_{-\infty}^{\infty} x f(x) , dx ), where ( f(x) ) is the probability density function.
   • It represents the average outcome over a continuous range of values.
   • The expected value can be interpreted as the "balance point" of the probability distribution.

3. Linearity of Expectation

   • The linearity of expectation states that the expected value of the sum of random variables is equal to the sum of their expected values, regardless of whether the variables are independent.
   • Mathematically, ( E(X + Y) = E(X) + E(Y) ).
   • This property simplifies calculations, especially when dealing with multiple random variables.

4. Expected Value of a Function of a Random Variable

   • If ( g(X) ) is a function of a random variable ( X ), the expected value can be calculated as ( E(g(X)) = \sum g(x_i) P(x_i) ) for discrete variables or ( E(g(X)) = \int g(x) f(x) , dx ) for continuous variables.
   • This allows for the analysis of transformed variables and their expected outcomes.
   • It highlights how functions can change the distribution and expected value of the original variable.

5. Conditional Expected Value

   • The conditional expected value ( E(X | Y) ) represents the expected value of a random variable ( X ) given that another random variable ( Y ) takes on a specific value.
   • It is calculated using the formula: ( E(X | Y = y) = \sum x_i P(X = x_i | Y = y) ) for discrete variables or ( E(X | Y = y) = \int x f_{X|Y}(x|y) , dx ) for continuous variables.
   • This concept is crucial for understanding dependencies between random variables.

6. Law of Total Expectation

   • The law states that the expected value of a random variable can be found by taking the expected value of its conditional expected values: ( E(X) = E(E(X | Y)) ).
   • It provides a way to break down complex problems into simpler components.
   • This law is particularly useful in scenarios involving multiple stages or layers of uncertainty.

7. Expected Value of a Binomial Distribution

   • For a binomial distribution with parameters ( n ) (number of trials) and ( p ) (probability of success), the expected value is given by ( E(X) = n \cdot p ).
   • This reflects the average number of successes in ( n ) independent Bernoulli trials.
   • The expected value helps in assessing the likelihood of outcomes in binomial experiments.

8. Expected Value of a Poisson Distribution

   • For a Poisson distribution with parameter ( \lambda ) (average rate of occurrence), the expected value is ( E(X) = \lambda ).
   • This indicates the average number of events occurring in a fixed interval of time or space.
   • The expected value is particularly useful in modeling rare events.

9. Expected Value of an Exponential Distribution

   • The expected value of an exponential distribution with rate parameter ( \lambda ) is given by ( E(X) = \frac{1}{\lambda} ).
   • This represents the average time until the next event occurs in a Poisson process.
   • The expected value is significant in reliability and survival analysis.

10. Expected Value of a Uniform Distribution

    • For a uniform distribution defined on the interval ([a, b]), the expected value is calculated as ( E(X) = \frac{a + b}{2} ).
    • This indicates that the average outcome is the midpoint of the interval.
    • The expected value reflects the equal likelihood of all outcomes within the specified range.