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. 1 / 12
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. 6 / 12
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
