STA301 - Module 4 (Exam 1)

expected value (µ or E[X]) of a random variable

the expected value of a random variable is its theoretical long-run average value it measures the center of the probability distribution it tells you what you would expect to see, on average, if you observed the random variable in a very large number of repetitions in the experiment (picture shows E[X] equation for a continuous random variable; see phone for equation for a discrete random variable)

Expected Value of Transformations of Random Variables

(see phone for equations for continuous and discrete random variables)

Chebyshev's rule

A Russian mathematician Chebyshev discovered a relationship between µ and σ for all random variables X. For any random variable X with finite σ² and for any k > 1: P(µ - kσ < X < µ + kσ) ≥ 1 - (1/k²) In other words, for any k > 1, at least (1 - (1/k²)) * 100% of the values of the distribution are within k standard deviations of µ. This rule should only be used when the distribution of the random variable is unknown. When the distribution of the random variable is known, we should use the PDF or PMF to find the probabilities because the bound given by Chebyshev's rule is often much different than the actual probability for the known distributions.

What is a distribution function? Name three types. What are the two ways that they are usually displayed?

A mathematical function that describes the behavior of a random variable is known as a distribution function. f(x) is the probability of x 1. Probability mass function (PMF) 2. Probability density function (PDF) 3. Cumulative distribution function (CDF) (there are more types than these three) distribution functions are usually displayed using either functional or tabular notation tabular notation consists of a table of the values of x and f(x) functional notation is a little different: Ex: Let X be a random variable for the number of spots that appear when you roll a six-sided die. f(x) = { ¹/₆, if x = 1, 2, 3, 4, 5, 6 { 0, elsewhere

What is a random variable? What notation is used to describe random variables?

A random variable is a function that simply assigns numerical value to each outcome in the sample space of a random experiment. Capital letters X, Y, Z, etc. are used to denote random variables. Lowercase letters x, y, z, etc. are used to denote possible values that the corresponding random variables can attain. Such as: let X be the number of electrons that flow through an electric current. After an experiment is conducted, the measured value of the random variable is denoted by a lowercase letter such as x = 70 milliamperes.

cumulative distribution function (CDF)

Another way to specify the distribution of probability is to assign probabilities to intervals of form (-∞, x] for all real values of x. The CDF of a random variable X, typically denoted by f(x), is defined as P(X ≤ x). (see Module 4 PowerPoint and Questions 12 and 13)

Law of Large Numbers

As the number of observations of a random variable increases, the average of the observations converges to the expected value. In other words, if we repeat an experiment a large number of times our observed average should be fairly close to the expected value.

Addition and subtraction rule for expected values of random variables: E(X ± Y) = ?

E(X ± Y) = E(X) ± E(Y)

Adding a constant c to a random variable X: E(X ± c) = ? Var(X ± c) = ? sd(X ± c) = ?

E(X ± c) = E(X) ± c Var(X ± c) = Var(X) sd(X ± c) = sd(X) Adding or subtracting a constant "c" from the random variable shifts the mean if c ≠ 0 but doesn't change the variance or standard deviation.

Multiplying a random variable X by a constant a: E(aX) = ? Var(aX) = ? sd(aX) = ?

E(aX) = aE(X) Var(aX) = a²Var(X) sd(aX) = |a|sd(X)

variance and standard deviation of a random variable

If X is a random variable, σ² = Var[X] = E[(x - µ)²]. The square root of the variance gives the standard deviation of a probability distribution. In other words, in a probability distribution, the standard deviation measures how far a set of possible values of a random variable is spread out on average from the population mean. (see phone for equations for continuous and discrete random variables)

continuous random variable

If a sample space contains an infinite number of possibilities corresponding to all the points in an interval of the real line, it is called a continuous random variable. Ex: electrical current, pressure, temperature, voltage, etc.

discrete random variable

If the set of all possible values of a random variable, X, is a countable set (finite or countably infinite), then X is called a discrete random variable. Ex: number of scratches on a surface, number of transmitted bits received in error

Addition and subtraction rules for variances and standard deviations of independent random variables: Var(X ± Y) = ? sd(X ± Y) = ?

Var(X ± Y) = Var(X) + Var(Y) sd(X ± Y) = √(Var(X) + Var(Y))

Bernoulli random variable

a special type of discrete random variable that can result in only two outcomes the outcomes are typically denoted as 0 for failure and 1 for success Ex: flip a coin once - heads, tails

What is a probability mass function (PMF)? What are the two conditions that need to be satisfied for a distribution function to be considered a valid PMF?

a type of distribution function f(x) = p(X = x) that assigns the probabilities to each possible value of the discrete random variable X 1. f(x↓i) ≥ 0 for all values of x 2. Σ f(x↓i) = 1 Ex: A small business just leased a new computer for three years. The computer company's research indicates that during a given year 86% of the computers need no repairs, 9% need to be repaired once, 4% twice, and 1% three times. Let X be a random variable for the number of computer repairs per year for the company. - this is a PMF because every value of f(x) is greater than 0 (0.86, 0.09, 0.04, 0.01) and the sum of all of the values of f(x) is equal to 1 (0.86 + 0.09 + 0.04 + 0.01 = 1)

What is a probability density function (PDF)? What are the two conditions that need to be satisfied for a distribution function to be considered a valid PDF?

a type of distribution function used to determine probabilities associated with continuous random variables 1. f(x) ≥ 0 for all values of x 2. ∫ f(x) dx = 1 from -∞ to ∞ (area under curve = 1)