Download Continuous Random Variables

ST 380
Probability and Statistics for the Physical Sciences
Continuous Random Variables
Recall:
A continuous random variable X satisfies:
1
its range is the union of one or more real number intervals;
2
P(X = c) = 0 for every c in the range of X .
Examples:
The depth of a lake at a randomly chosen location.
The pH of a random sample of effluent.
The precipitation on a randomly chosen day is not a continuous
random variable: its range is [0, ∞), and P(X = c) = 0 for any
c > 0, but P(X = 0) > 0.
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Continuous Random Variables
Probability Density Function
ST 380
Probability and Statistics for the Physical Sciences
Discretized Data
Suppose that we measure the depth of the lake, but round the depth
off to some unit.
The rounded value Y is a discrete random variable; we can display its
probability mass function as a bar graph, because each mass actually
represents an interval of values of X .
In R
source("discretize.R")
discretize(0.5)
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Continuous Random Variables
Probability Density Function
ST 380
Probability and Statistics for the Physical Sciences
As the rounding unit becomes smaller, the bar graph more accurately
represents the continuous distribution:
discretize(0.25)
discretize(0.1)
When the rounding unit is very small, the bar graph approximates a
smooth function:
discretize(0.01)
plot(f, from = 1, to = 5)
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Continuous Random Variables
Probability Density Function
ST 380
Probability and Statistics for the Physical Sciences
The probability that X is between two values a and b, P(a ≤ X ≤ b),
can be approximated by P(a ≤ Y ≤ b).
Because Y is discrete, P(a ≤ Y ≤ b) is the sum of the areas of the
corresponding bars in the graph.
As the rounding unit becomes smaller, the sum of the areas of the
bars approaches the integral of the smooth function.
In the limit,
Z
P(a ≤ X ≤ b) =
b
f (x) dx.
a
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Continuous Random Variables
Probability Density Function
ST 380
Probability and Statistics for the Physical Sciences
The smooth function f (x) is called a probability density function
(pdf).
Clearly f (x) must satisfy:
f (x) ≥ 0, −∞ < x < ∞;
Z ∞
f (x) dx = 1.
(1)
(2)
−∞
Any f (x) satisfying these two conditions could be the pdf of some
continuous random variable.
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Continuous Random Variables
Probability Density Function
ST 380
Probability and Statistics for the Physical Sciences
Uniform Distribution
A bee leaves its hive to forage for blossom that will provide nectar.
If the bee has no prior information, it searches in a random direction.
If X is the direction, measured from North clockwise in degrees, then
X is equally likely to be any value in [0, 360).
More precisely, if 0 ≤ a < b < 360, then
b−a
P(a ≤ X ≤ b) =
=
360
Z
a
b
1
dx.
360
So the pdf is f (x) = 1/360, 0 ≤ x < 360, and zero otherwise.
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Continuous Random Variables
Probability Density Function
ST 380
Probability and Statistics for the Physical Sciences
Cumulative Distribution Function
The cumulative distribution function F (x) of any random variable X
is defined as
F (x) = P(X ≤ x),
−∞ < x < ∞.
Earlier, for a continuous random variable X ,
Z b
P(a ≤ X ≤ b) =
f (y ) dy ,
a
so
Z
x
F (x) = P(X ≤ x) =
f (y ) dy .
−∞
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Continuous Random Variables
Cumulative Distribution Function
ST 380
Probability and Statistics for the Physical Sciences
Conversely,
f (x) =
dF (x)
= F 0 (x).
dx
For the uniform distribution on [0, 1),
(
1 0≤x <1
f (x) =
0 otherwise
and


0
F (x) =
f (y ) dy = x

−∞

1
Z
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x
x <0
0≤x <1
x ≥1
Continuous Random Variables
Cumulative Distribution Function
ST 380
Probability and Statistics for the Physical Sciences
Percentiles
The (100p)th percentile of X is the value that X falls below with
probability p.
That is, it is the value η(p) that satisfies
P[X ≤ η(p)] = p.
In terms of the cdf,
Z
η(p)
p = F [η(p)] =
f (y ) dy .
−∞
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Continuous Random Variables
Cumulative Distribution Function
ST 380
Probability and Statistics for the Physical Sciences
Median and Quartiles
Most commonly used percentiles:
The median is the 50th percentile,
1
P(X ≤ median) = .
2
The upper and lower quartiles are the 75th and 25th percentiles,
respectively,
3
P(X ≤ upper quartile) = ,
4
1
P(X ≤ lower quartile) = .
4
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Continuous Random Variables
Cumulative Distribution Function
ST 380
Probability and Statistics for the Physical Sciences
Expected Value
Recall that the expected value of a discrete random variable is the
average of its values, weighted by their probabilities.
For a continuous random variable, expected value is defined the same
way, but the average must be computed as an integral instead of a
sum:
Z ∞
µX = E (X ) =
x · f (x) dx.
−∞
More generally, for a function h(X ),
Z
µh(X ) = E [h(X )] =
∞
h(x) · f (x) dx.
−∞
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Continuous Random Variables
Expected Value
ST 380
Probability and Statistics for the Physical Sciences
Variance and Standard Deviation
As for a discrete random variable, the variance of X is
σX2 = V (X ) = E (X − µX )2
and the standard deviation is
σX =
p
V (X ).
But the expected value is now given by an integral:
Z ∞
2
(x − µX )2 · f (x) dx
E (X − µX ) =
−∞
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Continuous Random Variables
Expected Value