INTRODUCTORY STATISTICS
Chapter 5 CONTINUOUS RANDOM VARIABLES
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SEC. 5.2: CONTINUOUS PROBABILITY FUNCTIONS
The graph of a probability density function (abbreviated as pdf) is a curve. We use
the symbol f(x) to represent the curve.
Area under the curve is given by a different function called the cumulative
distribution function (abbreviated as cdf). The cumulative distribution function is used
to evaluate probability as area.
For continuous probability distributions, PROBABILITY = AREA.
UNIFORM DISTRIBUTION
In a pdf, the area under the curve represents probability, so the total area is 1. What is
the height of the box in this graph?
EXPONENTIAL DISTRIBUTION
NORMAL DISTRIBUTION
EXAMPLE:
a) What is f(x)?
𝑓 𝑥 =
1
, 𝑓𝑜𝑟 0 ≤ 𝑥 ≤ 20
20
EXAMPLE
Write the shaded area as a probability.
Calculate the probability.
EXAMPLE
Write the shaded area as a probability and calculate.
SEC. 5.3: THE UNIFORM DISTRIBUTION
The uniform distribution is a continuous probability distribution and
is concerned with events that are equally likely to occur.
Formulas for the theoretical mean and standard deviation are
𝜇=
𝑎+𝑏
2
and 𝜎 =
(𝑏−𝑎)2
12
EXAMPLE: THE CURRENT (IN MA) MEASURED IN A PIECE OF COPPER
WIRE IS KNOWN TO FOLLOW A UNIFORM DISTRIBUTION OVER THE
INTERVAL [0,25].
•
Fill in missing values on the x and f(x) axis if X = the current
•
Write the distribution of X
•
If the shaded region is half the graph, describe that region using probability notation.
CONTINUING THE COPPER EXAMPLE
•
Write the shaded region using probability notation and find the corresponding value.
•
Find the theoretical mean and standard deviation for the current.
•
Write the shaded region as a probability.
•
Find the theoretical mean
•
Find the theoretical standard deviation.
FINDING PERCENTILES
Find the value that represents the 90th percentile.
MORE PERCENTILES
Find the 30th percentile for this distribution.
ANOTHER PERCENTILE
Find the value that is the minimum amount for the 4th quartile (Q3)
•
Write the distribution of X.
•
Write the shaded area as a probability and find its value.
•
Find 𝑃 𝑥 < 15 𝑥 > 12
EXAMPLE
•
Write the distribution of X.
•
Write the shaded area as a probability and find its value.
•
Find 𝑃 𝑥 > 2 𝑥 < 3
EXAMPLE: IN AN ELEMENTARY SCHOOL, AGES OF
STUDENTS IS EVENLY DISTRIBUTED BETWEEN 3 AND 8
•
Define X and write its distribution.
•
Find the mean and standard deviation
•
Find P(4<x<7)
•
Find 𝑃 𝑥 < 6 𝑥 > 5
•
Find the 60th percentile of ages.
PRACTICE: WAIT TIMES AT THE DMV ARE
UNIFORMLY DISTRIBUTED FROM 5 TO 45 MINUTES.
• Define X and its distribution and sketch.
• Find the mean and standard deviation.
• Find P(x>20).
• Find 𝑃 𝑥 < 15 𝑥 < 40
• Find the 40th percentile of wait times.