Probability Density Function

7 Key excerpts on "Probability Density Function"

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  eBook - ePub

  Statistics for Process Control Engineers

  A Practical Approach

  • Myke King(Author)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  5 Probability Density Function

  The Probability Density Function (PDF) is a mathematical function that represents the distribution of a dataset. For example, if we were to throw a pair of unbiased six‐sided dice 36 times, we would expect (on average) the distribution of the total score to be that shown by Figure 5.1 . This shows the frequency distribution. To convert it to a probability distribution we divide each frequency by the total number of throws. For example, a total score of 5 would be expected to occur four times in 36 throws and so has a probability of 4/36 (about 0.111 or 11.1%). Figure 5.2 shows the resulting probability distribution.

  Figure 5.1

  Expected frequency of total score from two dice

  Figure 5.2

  Expected distribution

  Throwing dice generates a discrete distribution; in this case the result is restricted to integer values. Probability should not be plotted as continuous line. The probability of a non‐integer result is zero. But we can develop an equation for the line. In this case, if x is the total scored, the probability of scoring x is

  (5.1)

  Because x is discrete this function is known as the probability mass function (PMF). If the distribution were continuous we can convert the probability distribution to a Probability Density Function (PDF) by dividing the probability by the range over which it applies.

  A condition of both functions is that the area they contain must be unity (or 100%) – in other words we are certain to be inside the area. So, in general

  (5.2)

  Provided this condition is met then any function can be described as a PDF (or PMF). We will show later that there are many functions that have a practical application. Unfortunately there are a larger number that appear to have been invented as a mathematical exercise and are yet to be shown that they describe any real probability behaviour.

  While the PMF allows us to estimate the probability of x having a certain value, the PDF does not. It only tells us the probability of x falling within a specified range. The probability of x being between a and b

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  eBook - PDF

  Applied Statistics and Probability for Engineers

  • Douglas C. Montgomery, George C. Runger(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  But as in the discrete case, many physical systems can be modeled by the same or similar continuous random variables. These random variables are described, and example computations of probabilities, means, and variances are provided in the sections of this chapter. Density functions are commonly used in engineering to describe physical systems. For example, consider the density of a loading on a long, thin beam as shown in Figure 4.1. For any point x along the beam, the density can be described by a function (in grams/cm). Intervals with large loadings correspond to large values for the function. The total loading between points a and b is determined as the integral of the density function from a to b. This integral is the area under the density function over this interval, and it can be loosely interpreted as the sum of all the loadings over this interval. Similarly, a Probability Density Function f (x) can be used to describe the probability dis- tribution of a continuous random variable X. If an interval is likely to contain a value for X, its probability is large and it corresponds to large values for f (x). The probability that X is between a and b is determined as the integral of f (x) from a to b. See Figure 4.2. Loading x FIGURE 4.1 Density function of a loading on a long, thin beam. P(a < X < b) a b x f (x) FIGURE 4.2 Probability determined from the area under f (x). 68 CHAPTER 4 Continuous Random Variables and Probability Distributions Probability Density Function For a continuous random variable X, a Probability Density Function is a function such that (1) f (x) ≥ 0 (2) ∫ ∞ −∞ f (x) dx = 1 (3) P(a ≤ X ≤ b) = ∫ b a f (x) dx = area under f (x) from a to b for any a and b (4.1) A Probability Density Function provides a simple description of the probabilities associated with a random variable. As long as f (x) is nonnegative and ∫ ∞ −∞ f (x) dx = 1, 0 ≤ P(a < X < b) ≤ 1 so that the probabilities are properly restricted.

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  Probability with STEM Applications

  • Matthew A. Carlton, Jay L. Devore(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  Because for each histogram the total area of all rectangles equals 1, the total area under the smooth curve is also 1. The probability that the depth at a randomly chosen point is between any two numbers a and b is just the area under the smooth curve between a and b. It is exactly a smooth curve of the type pictured in Figure 5.1c that specifies a continuous probability distribution. DEFINITION A probability distribution or Probability Density Function (pdf) of a continuous rv X is a function f (x) such that for any two numbers a and b with a ≤ b, P(a ≤ X ≤ b) = ∫ b a f (x)dx 162 CONTINUOUS PROBABILITY DISTRIBUTIONS: GENERAL PROPERTIES That is, the probability that X takes on a value in the interval [a, b] is the area above this interval and under the graph of the density function, as illustrated in Figure 5.2. The graph of f (x) is often referred to as the density curve. FIGURE 5.2 P(a ≤ X ≤ b) = the Area Under the Density Curve Between a and b a b x f(x) The following two conditions must be satisfied for f (x) to be a legitimate pdf: 1. f (x) ≥ 0 for all x 2. ∫ ∞ −∞ f (x)dx = [area under the entire graph of f (x)] = 1 Analogous to the discrete case, the support of a pdf is defined as the set of all x values for which f (x) > 0. We will frequently display a pdf only for the values in its support, with the understanding that f (x) = 0 otherwise. EXAMPLE 5.4 The direction of an imperfection with respect to a reference line on a circular object such as a tire, brake rotor, or flywheel is often subject to uncertainty. Consider the reference line connecting the valve stem on a tire to the center point, and let X be the angle measured clockwise to the location of an imperfection. One possible pdf for X is f (x) = 1 360 0 ≤ x < 360 This pdf is graphed in Figure 5.3. Clearly, f (x) ≥ 0. The area under the density curve is just the area of a rectangle: (height)(base) = ( 1 360 ) (360) = 1.

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  Foundational and Applied Statistics for Biologists Using R

  • Ken A. Aho(Author)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  Because the density will be equal for all possible outcomes, and will depend only on the limits of the distribution, x is not required in the density function. If a random variable X has the pdf in Equation 3.7, we denote this as X ~ UNIF( a , b ). The distribution UNIF(2.5, 3) is shown in Figure 3.7. For any continuous pdf, the probability P ( a ≤ X ≤ x ) for ( x ≥ a ) can be calculated by inte-grating the pdf of X , while using a and x as the respective lower and upper limits of inte-gration. When a is the true lower limit of the pdf, then the result is the cdf. Thus, the continuous uniform cdf is F x b a dt a x ( ) . = -∫ 1 (3.8) EXAMPLE 3.4 Let X ~ UNIF(2.5, 3.0), then to find P (2.5 ≤ X ≤ 2.8), we have 1 1 3 2 5 2 2 2 5 2 8 2 5 2 8 2 5 2 8 2 5 2 8 b a dt dt dt t -= -= = = -∫ ∫ ∫ . . . . . . . . . . 2(2 8) 2(2 5) 6 . . . = 0 61 Probability Density Functions This agrees with the cdf depiction in Figure 3.7. We can check our calculus using R . integrand < -function(x) 2 integrate(Vectorize(integrand), 2.5, 2.8) 0.6 with absolute error < 6.7e-15 Note that we use the Vectorize function here since the integrand is simply a con-stant and not a function of x . Figure 3.7 clearly demonstrates that the continuous pdfs do not directly provide prob-ability. The total area under the pdf curve in Figure 3.7 equals one because it is a rect-angle whose consecutive sides have length 2 and 0.5, and 2 × 0.5 = 1. However, because b – a = 0.5, the density for any outcome is 2, and we know that a probability must be in the interval [0, 1]. 3.2.2.2 Normal Distribution The most commonly used continuous pdf in statistics is the normal distribution. * It is used to represent processes in which the most likely outcome is the average. The distribution is symmetric around the average. Thus, outcomes an identically increasing distance above and below the average, are identically and increasingly unlikely. The normal pdf is f x e x ( ) , = --      1 2 1 2 2 σ π μ σ (3.9) where 1.

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  Probability, Statistics, and Stochastic Processes for Engineers and Scientists

  • Aliakbar Montazer Haghighi, Indika Wickramasinghe(Authors)
  • 2020(Publication Date)
  • CRC Press

    (Publisher)

  Note 3.6 Since the set consisting of all subsets of is extremely large, it will be impossible to assign probabilities to all. It has been shown in the theory of probability that a smaller set, say B, may be chosen that contains all events of our interest. In this case, B is referred to as the Borel field , similar to what was defined for discrete random variables. The triplet ( Ω , B, P ) is called the probability space , where P is the probability (measure) of events. 93 Random Variables Example 3.8 Suppose the temperatures in Houston in the month of July in the past many years have always been between 95 °F and 105 °F . This means that the temperature can take any value between the ranges 94.5 °F and 105.5 °F . When we say that the tem-perature is 100 °F , it means that the temperature lies somewhere between 99.5 °F and 100.5 °F . This is an example of a continuous random variable. 3.2 DISCRETE PROBABILITY DISTRIBUTION (MASS) FUNCTIONS (PMF) A discrete random variable assigns discrete values to the sample points. As a ran-dom variable represents an event, we can assign probabilities to its values. A function describing how probabilities are distributed across the possible values of the random variable is referred to as the probability distribution of the random variable . In other words, in a random experiment, we might want to look at situations from which we can observe the predictable patterns and can use them as models. The patterns of such models are called the probability distributions . As a random variable may be discrete or continuous, so is its distribution. In this section, we discuss the discrete case. Note 3.7 A pmf of a random variable can be represented as a mathematical formula, table, or a graph. Definition 3.5 Let X be a discrete random variable (finite or countably infinite) defined on the sam-ple space Ω with range R x = { } 1 2 , x ,..., x n , where n is a nonnegative integer. Then, p X defined by p P X k = = { } X x , 1 k = ,2, ...

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  Statistical Methods Of Geophysical Data Processing

  • Vladimir Troyan, Yurii Kiselev(Authors)
  • 2010(Publication Date)
  • World Scientific

    (Publisher)

  To find a density function f ζ ( z ) for the random vari-able ζ = ξη . I n s t r u c t i o n . We fix some value z , then on a plane x 0 y we build a curve with an equation z = xy . It is a hyperbola. Distribution function is given by the formula F ζ ( z ) = P (( ξ, η ) ∈ D ) = P ( ξη < z ) , f ζ ( z ) = G 0 ζ ( z ) . (8) The system of random variables ( ξ, η ) has a joint density function f ξη ( x, y ). To find the density function f ζ ( z ) of their ratio ζ = η/ξ . (9) To find a distributive law ζ = η/ξ of two independent normally distributed variables ξ , η with parameters m ξ = m η = 0, σ ξ , σ η . (10) To find a distributive law of a random variable ζ = ξ + η , if ξ and η are independent random variables, which obey the exponential law with density functions f ξ ( x ) = λe -λx ( x > 0) , f η ( y ) = μe -μy ( y > 0) . Basic concepts of the probability theory 23 1.4 The Numerical Characteristics of Probability Distributions Let (Ω , A , P ) a finite probabilistic space and ξ = ξ ( ω ) is a random variable with the values belonging to a set X = { x 1 , . . . , x n } . If A i = { ω : ξ = x i } ( i = 1 , . . . , n ), then ξ is possible to represent as ξ ( ω ) = n X i =1 x i I ( A i ) , I ( A i ) = 1 ω ∈ A i , 0 ω 6 ∈ A i , where A 1 , . . . , A n is a dissection of the space Ω, i.e. pairwise are not intercrossed, and their sum is equal Ω. 1.4.1 Mathematical expectation Mathematical expectation, or ( expectation, expected value, mean of distribution ) of a random variable ξ = ∑ n i =1 x i I ( A i ), is named a value Mξ = n X i =1 x i P ( A i ) . To take into account A i = { ω : ξ ( ω ) = x i } and P ξ ( x i ) = P ( A i ), we can obtain Mξ = n X i =1 x i P ξ ( x i ) . (1.23) If the random variable ξ ∈ R 1 is continuous one, its mathematical expectation is defined as Mξ = ∞ Z -∞ xf ξ ( x ) dx. (1.24) The mathematical expectation is a measure of location of a random variable ξ and characterizes “center” of its distribution.

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  Random Vibrations

  Analysis of Structural and Mechanical Systems

  • Loren D. Lutes, Shahram Sarkani(Authors)
  • 2004(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  Thus, the random variables X and Y are said to have a continuous joint distribution. Clearly, for this particular problem the description given by the density function is much simpler than that given by the cumulative distribution function. ******************************************************************************************** Example 2.12: Let the joint Probability Density Function of X and Y be equal to the constant C on the set of possible values, as in Example 2.11, but let the space of possible values be a triangle in the ( , ) u v plane such that p u v C v u p u v XY XY ( , ) ( , ) = ≤ ≤ ≤ = for otherwise 1 5 0 A sketch of this joint Probability Density Function is shown. Find the value of the constant C and the joint cumulative distribution function F u v XY ( , ) for all values of ( , ) u v . Again we use total probability to find that C = 1 8 / , because the triangle of possible values has an area of 8 and the density function is constant on that triangle. Integration gives the joint cumulative distribution function as the continuous function p XY ( u , v ) u v v = u 1 1 5 5 Fundamentals of Probability and Random Variables 29 F u v u v v v u F u v u v u u F u v v v v u F u v u v F u v XY XY XY XY XY ( , ) ( )( ) / ( , ) ( ) / , ( , ) ( )( ) / , ( , ) , ( , ) = − − ≤ ≤ ≤ = − > ≤ ≤ = − − ≤ ≤ > = > > = 2 1 1 16 1 5 1 16 1 5 9 1 16 1 5 5 1 5 5 0 2 for for for for otherwise ******************************************************************************************** Example 2.13: Consider two random variables X and Y with the joint cumulative distribution given by F u v e e e e XY u v u v ( , ) ( )( ) ( )( ) = − − − −     − − − − 1 1 2 1 1 4 4 4 4 3 ( )( ) ( ) ( ) 1 1 4 3 4 − −     − − e e U u U v u v in which the unit step function has been used to convey the information that the nonzero function applies only in the first quadrant of the ( , ) u v plane. Find the joint Probability Density Function for X and Y .

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