Stats 1 week 10

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Summary

Random Variable

A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. There are two types:

• Discrete Random Variables: Take on a countable number of distinct values (e.g., the number of heads in 10 coin flips).
• Continuous Random Variables: Take on an infinite number of possible values (e.g., the exact height of students in a class).

Expectation of a Random Variable

The expectation (or expected value) of a random variable is a measure of the center of its distribution. For a discrete random variable X with possible values x₁, x₂, ..., x_n and probabilities p₁, p₂, ..., p_n, the expectation is given by:

E(X) = ∑_i=1ⁿ x_i p_i

Failure and Success

In the context of probability and statistics, failure and success refer to the two possible outcomes of a Bernoulli trial. A success is typically assigned a value of 1, and a failure is assigned a value of 0.

Discrete Uniform Random Variable

A discrete uniform random variable is one that has a finite number of outcomes, each of which is equally likely. If X is a discrete uniform random variable with n possible outcomes, then each outcome has a probability of 1/n.

Bernoulli Random Variable

A Bernoulli random variable is a discrete random variable that takes the value 1 with probability p (success) and the value 0 with probability 1-p (failure). It is used to model a single trial of a binary experiment.

Expectation of a Bernoulli Random Variable

For a Bernoulli random variable X with parameter p:

E(X) = p

Variance

The variance of a random variable measures the spread of its values around the mean. For a discrete random variable X with mean E(X), the variance is given by:

Var(X) = E[(X - E(X))²]

Multiplying a Constant with Expectation and Variance

Expectation: If X is a random variable and a is a constant, then:

E(aX) = aE(X)

Variance: If X is a random variable and a is a constant, then:

Var(aX) = a² Var(X)

Adding a Constant to Expectation and Variance

Expectation: If X is a random variable and b is a constant, then:

E(X + b) = E(X) + b

Variance: If X is a random variable and b is a constant, then:

Var(X + b) = Var(X)

Expectation and Variance of Two Random Variables

Expectation: If X and Y are random variables, then:

E(X + Y) = E(X) + E(Y)

Variance: If X and Y are independent random variables, then:

Var(X + Y) = Var(X) + Var(Y)