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Lecture 13

# STAT 2060 Lecture Notes - Lecture 13: Standard Deviation, Probability Distribution, Random Variable

Department
Statistics
Course Code
STAT 2060
Professor
Peter Kim
Lecture
13

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Homework Questions: Custom edition, Table II p 797 or 11th edition, Table IV p 822 or 10th
edition, Table IV p 884 → Custom edition, p. 233: 4.84–4.89. or 11th edition, p. 216: 4.79–4.85.
or 10th edition, p. 240: 4.84–4.90 → Custom edition, p. 233: 4.91–4.96 or 11th edition, p. 216:
4.87–4.90 or 10th edition, p.240: 4.91–4.96
STAT2060: Week 5
Random Variables II: Continuous
Preliminaries
oA random variable is the realization of what we previously called an experiment.
oA random variable can take on discrete values in which it will be called a discrete
random variable.
oA random variable can take on continuous values in which it will be called a
continuous random variable.
oWe often denote by x, or X as the random variable which can be either discrete
or continuous depending on context.
Examples of continuous random variables
oThe foreign exchange rate over a trading day, x ≥ 0.
oThe price of Apple stocks over a trading day, x ≥ 0.
oThe time it takes to log onto a popular website, x ≥ 0.
Continuous probabilities
oThe probability density function for a continuous random variable x consists of
function with the following properties:
f (x) ≥ 0 for all x
P(a < x < b) is given as an area under the f (x)
Things to note
oThe continuity of f (x) implies that P(x = a) = 0. This is because there is no “area”
at a single point.
o This means that unlike discrete random variables, for continuous random
variables there is no ambiguity associated with including endpoints
P(a < x < b) = P(a ≤ x ≤ b).
Population distribution for a normal random variable
oThere are many different types of continuous random variables, but the most
famous and practically useful one is the normal or Gaussian distribution named
after Carl F. Gauss (1777-1855).
oThe probability density function is: