STATS 311 - Chapter 8

Binomial Experiment

A binomial experiment is defined by the following conditions: 1. There are n "trials," where n is specified in advance and is not a random value. 2. There are two possible outcomes on each trial. The outcomes are called "success" and "failure" and denoted S and F. 3. The outcomes are independent from one trial to the next. 4. The probability of a "success" remains the same from one trial to the next, and this probability is denoted by p. The probability of a "failure" is 1-p for every trial.

Discrete Random Variable

A discrete random variable can take one of a countable list of distinct values. For discrete random variables we can find probabilities for exact outcomes. An example of a discrete random variable is the number of people with type O blood in a sample of ten individuals. The possible values are , a list of distinct values.

Family of Random Variables

A family of random variables consists of all random variables for which the same formula is used to find probabilities. In considering random variables, the first step is to identify whether the random variable fits into any known family. Then, the formulas for that family can be used to find probabilities for the possible outcomes. Discrete or Continuous Variables.

Normal Distribution

A normal random variable is said to have a "normal distribution."

Standard Normal Random Variable

A normal random variable with mean μ = 0 and standard deviation σ = 1 is said to be a standard normal random variable.

Standard Normal Distribution

A normal random variable with mean μ = 0 and standard deviation σ = 1 is said to have a standard normal distribution. The standard normal distribution is the distribution of standardized scores (Z-Scores) for a normal random variable

Random Variable

A random variable assigns a number to each outcome of a random circumstance. Equivalently, a random variable assigns a number to each unit in a population.

Uniform Random Variable

A random variable with this property is called a uniform random variable and is the simplest example of a continuous random variable. For this type of variable, the probability for an interval equals the area of a rectangle. If the range of a uniform random variable is from the number a to the number b, then the probability density function has a height of 1/(b-a) for all values from a to b, and a height of 0 everywhere else.

Normal Curve

A specific form of a bell-shaped probability density curve.

Independent Random Variables

For a linear combination of independent random variables, Variance (L) is: a²Variance(X) + b² Variance(Y) +.... A linear combination of independent normal random variables has a normal distribution.

Z-Score

Standardized Score. The distance between a specified value and the mean, measured in number of standard deviations. z = (Value - Mean)/(Standard Deviation) = (x - μ)/σ

Binomial Random Variable

A binomial random variable is X = number of successes in "n" independent trials of a random circumstance in which p = probability of success is the same in each trial.

Bernoulli Random Variable

Any individual random circumstance can be treated as a binomial experiment with n=1 and p=probability of a particular outcome. In this case, the value of the random variable X is either 0 or 1, and it may also be called a Bernoulli random variable. Example: ~ Random Circumstance: Roll two dice once ~ Random Variable: X= number of times sum is 7 ~ Success: Sum is 7 ~ Failure: Sum is not 7

Mean for a Population

E(X) = Σk(j)/N = Σx(i)p(i)

Variance for Binomial

For a binomial random variable, the variance is V(X) = np(1-p)

Mean Value of a Discrete Random Variable

If we know probabilities for all possible values of a random variable, we can determine the mean outcome over the long run. This long-run average is called the expected value of the random variable. Synonymously, it is the mean value of the random variable.

Statistically Independent

Two random variables are statistically independent if the probability for any event associated with one random variable is not altered by whether or not any particular event for the other random variable has happened. In a more practical sense, it is usually safe to say that two random variables are independent if there is no physical connection between the two variables or if there is no apparent reason why the value of one variable should influence the value of the other.

Variance of a Discrete Random Variable

Variance of X = σ² = Σ(x(i) - μ)^2 * p(i)

Standardized Score

Z-Score The standard normal distribution is the distribution of standardized scores for a normal random variable

Continuous Random Variable

A continuous random variable can take any value in an interval or collection of intervals. For continuous variables we are limited to finding probabilities for intervals of values. An example of a continuous random variable is height for adult women. With accurate measurement to any number of decimal places, any height is possible within the range of possibilities for heights. Between any two heights, there always are other possible heights, so possible heights fall on an infinite continuum.

Cumulative Probability

A cumulative probability is the probability that the value of a random variable is less than or equal to a specific value. A few examples follow: ~ Probability of getting or fewer answers right when guessing at answers to ten true-false questions ~ Probability that there are or fewer televisions in a randomly selected household The notation for a cumulative probability is P(X ≤ k), where X represents the random variable and k is a specific value. The term cumulative is used because P(X ≤ k) tells us how much probability X has accumulated for all values up to and including the value k.

Difference of Random Variables

For two random variables X and Y, the most commonly encountered linear combinations are the sum X+Y and the difference X-Y.

Sum of Random Variables

For two random variables X and Y, the most commonly encountered linear combinations are the sum X+Y and the difference X-Y.

Linear Combination of Random Variables

In a linear combination, we add (or subtract) variables, but some of the combined variables may be multiplied by a numerical value, as occurred in calculating the final score. A linear combination of random variables X,Y, ... had the form L=aX +bY + ... For a linear combination L=aX +bY + ..., Mean (L) is: aMean(X) + bMean(Y) +....

Standard Deviation for a Population

Standard Deviation of X = σ = σ = sqrt((Σ(x(i) - μ)^2)/N) = sqrt(Σ(x(i) - μ)^2 * p(i))

Cumulative Distribution Function (CDF)

The cumulative distribution function (cdf) for a random variable X is a table or rule that provides the probabilities P(X ≤ k) for any real number k.

Percentile Rank

The cumulative probability for the value of a variable. (The area to the left under the density curve for that value.)

Expected Value

The expected value of a random variable X is the mean value of the variable in the sample space or population of possible outcomes. Expected value can also be interpreted as the mean value that would be obtained from an infinite number of observations of the random variable. Expected Value = Sum of "value x probability," summed over all possible values

Mean for Binomial

The mean value for a binomial random variable is μ=np

Normal Random Variable

The most commonly encountered type of continuous random variable. A normal random variable has a symmetric, bell-shaped distribution referred to as the normal distribution.

Normal Approximation to the Binomial Distribution

The normal approximation to the binomial distribution is based on the following result, derived mathematically. If "X" is a binomial random variable based on "n" trials with success probability "p," and "n" is sufficiently large, then "X" is also approximately a normal random variable. The mean and standard deviation for this normal random variable are the same as for the binomial random variable: Mean = μ = np Standard Deviation = σ = sqrt(np(1-p)) Conditions: The approximation works well when both np and n(1-p) are at least 10.

Probability Density Function

The probability density function for a continuous random variable X is a curve such that the area under the curve over an interval equals the probability that X is in that interval. In other words, the probability that X is between the values a and b is the area under the density curve over the interval between the values a and b. The area under the curve for the entire range of possible values must equal 1.

Probability Distribution Function (PDF)

The probability distribution function (pdf) for a discrete random variable X is a table or rule that assigns probabilities to the possible values of the random variable X. (Note that the word function is used to mean either a table of values and probabilities or a formula (rule) that assigns probabilities to values.) P(X=x(i)) = 1

Binomial Distribution

The probability distribution function for a binomial random variable is called a binomial distribution.

Standard Deviation for Binomial

The standard deviation for a binomial random variable is σ = sqrt(np(1-p))

Standard Deviation of a Discrete Random Variable

The standard deviation of a discrete random variable quantifies how spread out the possible values of a discrete random variable might be, weighted by how likely each value is to occur. Variance of X = square root of V(X) = σ = sqrt(Σ(x(i) - μ)^2 * p(i))

Percentile

The value of the variable.

Continuity Correction

To make an approximation more accurate, sometimes a continuity correction is used by adding or subtracting , based on which rectangles are desired. For example, we could use the continuity correction when we find P(X ≤ 240) so that we would find P(X ≤ 240.5) instead, which would thus give us a more accurate result.