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Continuous Probability Functions
Continuous random variable - A variable whose value results from the measurement of a quantity that is subject to variations due to chance and which expresses a measurement as opposed to a discrete countable item.
Probability Density Function (PDF) - In a discrete model, a function that assigns a probability to each value. In a continuous model, the function assigns probability to the area under the curve in an interval. Probability functions must follow the Rules of Probability.
Cumulative Distribution Function (CDF) - A function that shows the cumulative sum of probabilities in a distribution.
Integral - Expresses an area under a probability curve.
Uniform Continuous Distribution - A distribution in which a random variable can be assumed to be uniformly distributed between two given points, a and b.
Normal/Gaussian Distribution - Always symmetric. All normal distributions have the same shape. The normal distribution is defined by two parameters and is denoted N(µ, σ). The interval [µ - 3σ, µ + 3σ] include almost all the area (following the Empirical Rule) of a normal distribution.
Standard Normal Distribution - A normal distribution with variables that have been transformed to universal application or "standardized". This is done by subtracting the mean and dividing by the standard deviation of the distribution.
7.1 Describing a Continuous Distribution
A discrete random variable is usually used for counting things. A continuous random variable is used for the measurement of things. These variables can be in non-integer form, such as using decimals and/or fractions of a particular time, weight, volume, etc. Therefore, the continuous variable cannot be expressed as being "at" a certain value of X. There would be infinite divisions between that value and the next. The continuous distribution would have to be expressed in terms of a smooth curve and the probabilities contained therein, would have to be expressed in terms of areas under the curve.
A continuous distribution can be described either by its PDF or CDF. For any continuous random variable, the PDF is an equation that shows the height of the curve f(x) (stated as "f of x" or "function of x") at each possible value of X. Any continuous PDF must be nonnegative and the area under the entire PDF must be 1. The CDF is denoted, F(x) and shows P(X ≤ x), the cumulative area to the left of a given value of X.
Since continuous probability functions are smooth curves, the area at any point would be zero. Because P(X = a) = 0, the expression P(a < X < B) is equal to P(a ≤ X ≤ b).
Expected Value and Variance
The mean and variance of a continuous random variable are analogous to E(X) and V(X) for a discrete random variable except that the integral sign ∫ replaces the summation sign ∑. Integrals are taken over all X-values.
7.2 Characteristics of the Uniform Distribution
Perhaps the simplest model one can imagine. If X is a random variable that is uniformly distributed between points a and b, its PDF will have a constant height. Sometimes denoted U(a, b).
Since its PDF will be rectangular, one can easily verify that the area sums to 1 by multiplying its base (b - a) by its height 1/(b - a).
Detail of Characteristics
Parameters
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a = lower limit; b = upper limit
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PDF
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f(x) = 1/b-a
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Mean
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a+b/2
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Standard Deviation
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√(b-a)2/12
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Comments
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Used as a conservative “what-if” benchmark
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Uses of the Uniform Model
The uniform model is used only when you have no reason to imagine that any X values are more likely than others. It's not a common experience in reality, but it can be useful in hypothetical analyses when one knows the "worst" and "best" range, but can't make any assumptions about values in the middle.
7.3 Characteristics of the Normal Distribution
A normal probability distribution is defined by two parameters, the mean, µ, and the standard deviation, σ. It is often denoted N(µ, σ). Its domain of a normal random variable is -∞ < x < +∞, but the application of the Empirical Rule allows the interval of [µ - 3σ, µ + 3σ] to be expressed as including nearly all values in a normal distribution. The expected value of a normal random value is the mean µ and that its variance will be the square of the standard deviation σ2. It is always symmetric. All normal distributions have the same shape, differing only in the axis scales.
Normal random variables can be found in economic and financial data. The normal distribution is especially important for a sampling distribution and for hypothesis testing. To qualify as "normal", a random variable should:
· Be measured on a continuous scale
· Possess a clear central tendency
· Have only one peak
· Exhibit tapering tails
· Be symmetric about the mean.
When the range is large, we often treat a discrete variable as continuous.
Ex. Exam scores are discrete (range from 0 to 100) but are often treated as continuous data. Here are some other variables that might be expected to be normally distributed:
X = quantity of beverage in a 2-liter bottle of Diet Pepsi
X = cockpit noise level in a Boeing 777 at the captain’s left ear during cruise
X = diameter in millimeters of a manufactured steel ball bearing
Each of these variables would tend toward a certain mean, but would exhibit random variation. For example, even with excellent quality control, not every bottle of soft drink will have exactly the same fill (even if the variation is only in a few milliliters). The mean and standard deviation will depend on the data generating process. Precision manufacturing can achieve a very small σ (standard deviation) in relation to µ (the mean), while other data generating situations can produce relatively large σ in relation to µ (e.g. your driving fuel mileage). Therefore, even if normal distributions have similar shapes, they can have different coefficients of variation.
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