STATS 250 1st Edition Lecture 5 Outline of Last Lecture I. Types of Research Studies and VariablesII. Interpretations of ProbabilityIII. Basic Rules of Finding ProbabilityOutline of Current Lecture I. What is a Random Variable?II. General Discrete Random VariablesIII. Expectations for Random VariablesIV. Binomial Random VariablesV. General Continuous Random VariablesVI. Normal Random VariablesCurrent LectureI. What is a Random Variable?a. Random variable: assigns a number to each outcome of a random circumstance, or, equivalently, a random variable assigns a number to each unit in a populationi. Discrete Random Variable: can take one of a countable list of distinct values1. Binomial: arises when counting the number of successes in a sampleii. Continuous Random Variable: can take any value in an interval or collection of intervals1. Uniform2. Normalb. Random variables are denoted by capital lettersc. Outcomes of random variables are denoted with lower case lettersII. General Discrete Random Variablesa. Discrete Random Variable: variable with a finite or countable number of possible outcomesb. Probability of variable X is equal to k is denoted at P(X=k)These notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.c. Probability Distribution Function (PDF) of a discrete random variable is a table/rule that assigns probabilities to the possible values of Xi. Sum of all individual probabilities must = 1ii. Individual probabilities must be between 0 and 1iii. Displayed in a probability histogram, or probability stick graph1. X-axis: values/outcomes2. Y-axis: probabilities of outcomesiv. Cumulative Distribution Function (CDF) of a discrete random variable is a table/rule that provides the probabilities P(X=k) for any real number k, where X is less than or equal to a certain value III. Expectations for Random Variablesa. Expected Value: mean value of the variable X in the sample space, or population, of possible outcomesi. Denoted E(X)ii. Can also be interpreted as the mean value that would be obtained from an infinite number of observations on the random variableiii. E(X) = sum (xi*pi)iv. Variance, V(X) = sum ((xi-µ)2*pi)v. Standard Deviation = square root of V(X)IV. Binomial Random Variablesa. Counts the number of times a certain event occurs out of a particular number of observations of trials of a random experimentb. Conditionsi. There are n trials, n determined in advance and not a random valueii. There are two possible outcomes of each trial: success (S) and failure (F)iii. The outcomes are independent from one trial to the nextiv. The probability of a success remains the same from one trial to the next, and this probability is denoted by pv. The probability of a failure is 1 – p for every trialc. Mean of Binomial Distribution: npi. Standard deviation = square root of (np(1-p))d. Binomial Formula for k = 0, 1, 2…V. General Continuous Random Variablesa. Continuous Random Variable: takes on all possible values in an intervalb. Described by a probability density curvei. Lies above the horizontal axisii. Total area under the curve = 1c. Mean of an expected variable = µVI. Normal Random Variablesa. If a population of measurements follows a normal curve, then X is a normal random variable and has a normal distributionb. Normal curve: symmetrical, bell-shaped, centered at the mean, µ determined bystandard deviationc. Standardized Score = (value-mean)/(standard