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Chapter 4

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Western University

Statistical Sciences

Statistical Sciences 2244A/B

Jennifer Waugh

Spring

Description

42 Random Variables
● Random variable: a variable that has a single numerical value, determined by chance for
each outcome of a procedure
○ Discrete random variable: either has finite number of values or countable number
of values where “countable” means there might be infinitely many values but can
be associated with counting process
○ Continuous random variable: infinitely many values and they can be associated
with measurements of a continuous scale without gaps or interruptions
● Probability distribution: graph, table, or formula that gives probability for each value of
the random variable
Graphs
● Probability Histogram: Is similar to the frequency histograms, but instead the y-axis
shows probabilities instead of relative frequencies based on actual sample results
● The areas of the rectangles on a probability histogram are the same as the probabilities
● Requirements for a Probability Distribution:
○ Sum of P(x) = 1 (sum of all probabilities is equal to 1)
○ 0 <= P(x) <= 1 (each probability value must be between 0 and 1 inclusive)
Identifying Unusual Results with Probabilities
● Remember, when it comes to calculating probabilities for events, we must try to
determine whether or not a given assumption is correct or not depending on if the
probability is very small or not (rare event rule)
● We need to be able to determine if events occurred due to chance or because a certain
technique is effective in providing certain results
● Agood example to keep in mind is flipping a coin 1000 times and getting 501 heads
● Although this seems like a normal result, as the probability of getting heads or tails every
time you flip is 0.5, thus getting at least 501 heads has a reasonable probability, the
probability of getting exactly 501 heads is much smaller (0.0252)
● Despite the small probability, we do not consider 501 heads out of 1000 unusual, because
getting at least 501 heads results in a probability of 0.487 which is high and reasonable
in this case
● Unusually high: x successes among n trials is an unusually high number of successes if
P(x or more) is very small (0.05 or less)
● Usually low: x successes among n trials is unusually low number of successes if P(x or
fewer) is very small (0.05 or less)
Expected Value
● The mean of a discrete random variable is the theoretical mean outcome for infinitely
many trials
● We can think of that mean as the expected value in the sense that it is the average value
that we would expect to get if the trials could continue indefinitely
● Expected value: of a discrete random variable is denoted by E, and represents average
value of outcomes, obtained by finding value of sum of x * P(x)
○ E is therefore equal to the population mean of a discrete random variable

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