Cumulative Distribution Function
A Cumulative Distribution Function (CDF) is a function that gives the probability that a random variable is less than or equal to a certain value. It provides a complete description of the probability distribution of a random variable. The CDF is a fundamental concept in probability theory and statistics, and it is used to analyze and interpret data.
8 Key excerpts on "Cumulative Distribution Function"
eBook - ePub- Ayanendranath Basu, Srabashi Basu(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
...In practice, it may also be characterized by the Cumulative Distribution Function (or simply distribution function) F (x) of the random variable. For each x in the support of X, this function is defined as F (x) = P r [ X ≤ x ]. It is easily seen that this function may be constructed from the probability mass function f (x). If the possible values of X are listed in an increasing order as x 1 < x 2 < …, then we have the relation F (x) = ∑ x i ≤ x f (x i). Also, F (x) = 0 for any x < x 1. If the largest value in the support of X is finite, then F (x) = 1 for any x equal to or greater than it. Also, for any i > 1, we have f (x i) = F (x i) − F (x i − 1), while for i = 1 we have f (x 1) = F (x 1). Thus one can construct the Cumulative Distribution Function (CDF) from the PMF by aggregating from the left, while one can also recover the PMF from the CDF by taking successive differences. Example 6.12. Continuing with Example 6.4, consider the PMF considered therein and a random variable having such a PMF. By cumulating the successive probabilities, we see that the CDF corresponding to this random variable is given as F (x) = 0 for x < 0, F (x) = 1 / 2 for 0 ≤ x < 1, F (x) = 3 / 4 for 1 ≤ x < 2, F (x) = 1 for x ≥ 2. A graph of this CDF is presented in Figure 6.2. Note also that the CDF given in Figure 6.2 has the nature of a step function, which is the general characteristic of the CDF of any discrete random variable. The graph has jumps at the support points and stays constant between the support points. There are some other properties which every CDF must satisfy (including the CDFs of continuous distributions to be defined in the next chapter). These properties are important, although to some extent technical. We discuss these properties in the appendix, with particular reference to Example 6.12...
eBook - ePubStatistical Power Analysis for the Social and Behavioral Sciences
Basic and Advanced Techniques
- Xiaofeng Steven Liu(Author)
- 2013(Publication Date)
- Routledge(Publisher)
...Appendix A Cumulative Distribution Function The Cumulative Distribution Function (cdf) for central t, F, or χ 2 can be viewed as a special case of non-central t, F, or χ 2 because the central t, F, or χ 2 assumes a zero non-centrality parameter. We only need to explain the cdf for the non-central t, F, or χ 2 to cover both the central and non-central cdf functions. We will first use a normal distribution to introduce the concept of the cdf, whose inverse function is a quantile. We will then express all the non-central cdf functions in a simple and unified way. A.1 Cumulative Distribution Function and Quantile The histogram in Figure A.1 shows the proportions of different values over the range of an arbitrary random variable X in a sample. For illustration, we can let X be a normal variable. When the sample size increases to the size of the population, the histogram on the left of Figure A.1 becomes the population distribution of X on the right. We denote any realized value of X as x. The Cumulative Distribution Function (cdf) is the probability of obtaining any realized value smaller than or equal to an arbitrary value x, namely, P[X ≤ x ], which is represented by the shaded area to the left of X under the distribution curve in Figure A.1. The cdf evaluates to p = P[X ≤ x ]. The inverse of the cdf function p is the quantile X = quantile (p). Therefore, X is called the pth quantile of X. The illustration assumes a normal variable X, although the concept of cdf and quantile applies to any other random variable, say, t, F, or χ 2. Figure A.1 Histogram and probability distribution A.2 Non-central Cumulative Distribution Function The current algorithms for computing the non-central cdfs involve different series of expansion and recurrence. These complicated algorithms can be greatly simplified. The three non-central cdfs can all be expressed as the integral of the normal cdf and the chi square density function...
eBook - ePub- John D. Holmes, Seifu Bekele(Authors)
- 2020(Publication Date)
- CRC Press(Publisher)
...Thus, the probability that X lies between a and b is: Pr { a < X < b } = ∫ a b f X (x) d x (C.1) Since any value of X must lie between − ∞ and + ∞ : ∫ − ∞ ∞ f X (x) d x = Pr { − ∞ < X < ∞ } = 1 Thus, the area under the graph of f X (x) versus x must equal 1.0. C.2.2 Cumulative Distribution Function (c.d.f.) The Cumulative Distribution Function F X (x) is the integral between. − ∞ and x of f X (x). i.e. F X (x) = ∫ − ∞ x f X (x) d x = Pr { − ∞ < X < x } = Pr { X < x } (C.2) The complementary Cumulative Distribution Function, usually denoted by G x (x) is: G X (x) = 1 − F X (x) = Pr { X > x } (C.3) F X (a). and G X (b) are equal to the areas indicated on Figure C.2. Figure C.2 Probability density function and Cumulative Distribution Functions. Note that: f X (x) = d F X (x) d x = − d G X (x) d x. (C.4) C2.3 Moments of the p.d.f. The following basic statistical properties of a random variable are defined, and their relationship to the underlying probability distribution is given. Mean X ¯ = (1 / N) ∑ i x i = ∫ − ∞ ∞ x f X (x) d x (C.5) Thus, the mean value is the first moment of the probability. density function (i.e. the x coordinate of the centroid of the area under the graph of the p.d.f.) Variance and standard deviation Variance: σ X 2 = (1 / N) ∑ i [ x i − X ¯ ] 2 (C.6) σ X (the square root of the variance) is called the standard deviation σ X 2 = ∫ − ∞ ∞ (x − X ¯) 2 f x (x) d x (C.7) Thus, the variance is the second moment of the p.d.f. about the mean value. It is analogous to the second moment of area of a cross-section about a centroid. Skewness s X = [ 1 / (σ X 3) ] Σ i [ x − X ¯ ] 3 = (1 / σ X 3) ∫ − ∞ ∞ (x − X ¯) 3 f X (x) d x (C.8) The skewness is the normalised third moment of the probability density function. Positive and negative skewness are illustrated in Figure C.3...
eBook - ePub- Maria L. Rizzo(Author)
- 2019(Publication Date)
- Chapman and Hall/CRC(Publisher)
...Chapter 2 Probability and Statistics Review In this chapter we briefly review without proofs some definitions and concepts in probability and statistics. Many introductory and more advanced texts can be recommended for review and reference. On introductory probability see e.g., Bean [ 26 ], Ghahramani [ 124 ], or Ross [ 249 ]. Mathematical statistics and probability books at an advanced undergraduate or first-year graduate level include, e.g., DeGroot and Schervish [ 69 ], Freund (Miller and Miller) [ 209 ], Hogg, McKean and Craig [ 150 ] or Larsen and Marx [ 177 ]. Casella and Berger [ 40 ] or Bain and Englehart [ 21 ] are somewhat more advanced. Durrett [ 80 ] is a graduate probability text. Lehmann [ 180 ] and Lehmann and Casella [ 181 ] are graduate texts in statistical inference. 2.1 Random Variables and Probability Distribution and Density Functions The Cumulative Distribution Function (cdf) of a random variable X is F X defined by F X (x) = P (X ≤ x), x ∈ ℝ. In this book P (·) denotes the probability of its argument. We will omit the subscript X and write F (x) if it is clear in context. The cdf has the following properties: 1. F X is non-decreasing. 2. F X is right-continuous; that is, lim ϵ → 0 + F X (x + ϵ) = F X (x), for all x ∈ ℝ. 3. lim x → − ∞ F X (x) = 0 and lim x → ∞ F X (x) = 1. A random variable X is continuous if F X is a continuous function. A random variable X is discrete if F X is a step function. Discrete distributions can be specified by the probability mass function (pmf) p X (x) = P (X = x). The discontinuities in the cdf are at the points where the pmf is positive, and p (x) = F (x) − F (x −). If X is discrete, the cdf of X is F X (x) = P (X ≤ x) = ∑ { k ≤ x : p X (k) > 0 } p X (k). Continuous distributions do not have positive probability mass at any single point...
eBook - ePub- Byron J.T. Morgan(Author)
- 2018(Publication Date)
- Chapman and Hall/CRC(Publisher)
...In detail here, the sample-space contains just two outcomes, say ω 1 and ω 2, corresponding to arrival and departure, respectively, and we can write the random variable X as X (ω), so that Χ (ω 1) = +1 and Χ (ω 2) = – 1. However, we find it more convenient to suppress the argument of X (ω), and simply write the random variable as X in this example. We shall adopt the now standard practice of using capital letters for random variables, and small letters for values they may take. When a random variable X is simulated n times then we obtain a succession of values: { x 1, x 2, …, x n }, each of which provides us with a realization of X. Another random experiment results if we record the queue size in the model of Exercise 1.4. From Section 1.6 we see that if p < 1 2, then after a long period of time since the start of the queueing system we can denote the queue size by a random variable, Y, say, such that Y may take any non-negative integral value, and Pr(Y = k) for k ≥ 0 is as given in Section 1.6. 2.2 The Cumulative Distribution Function (c.d.f.) For any random variable X, the function F, given by F (x) = Pr(X ≤ x) is called the Cumulative Distribution Function of X. We have lim x → ∞ F (x) = 1 ; lim x → − ∞ F (x) = 0 F (x) is a nondecreasing function of x,. and F (x) is continuous from the right (i.e. if x > x 0, lim x → x 0 F (x) = F (x 0)). The nature of F (x) determines the type of random variable in question, and we shall normally specify random variables by defining their distribution, which in turn provides us with F (x). If F (x) is a step function we say that X is a discrete random variable, while if F (x) is a continuous function of x then we say that X is a continuous random variable...
eBook - ePubProbability, Statistics, and Data
A Fresh Approach Using R
- Darrin Speegle, Bryan Clair(Authors)
- 2021(Publication Date)
- Chapman and Hall/CRC(Publisher)
...4 Continuous Random Variables DOI: 10.1201/9781003004899-4 Continuous random variables are used to model random variables that can take on any value in an interval, either finite or infinite. Examples include the height of a randomly selected human or the error in measurement when measuring the height of a human. We will see that continuous random variables behave similarly to discrete random variables, except that we need to replace sums of the probability mass function with integrals of the analogous probability density function. 4.1 Probability density functions Definition 4.1. A probability density function (pdf) is a function f such that: f (x) ≥ 0 for all x. ∫ f (x) d x = 1. Example 4.1. Let f (x) = 2 x 0 ≤ x ≤ 1 0 otherwise We see that f (x) ≥ 0 by inspection. and ∫ − ∞ ∞ f (x) d x = ∫ 0 1 2 x d x = 1, so f is a probability density function. Definition 4.2. A continuous random variable X is a random variable described by a probability density function, in the sense that: P (a ≤ X ≤ b) = ∫ a b f (x) d x. whenever a ≤ b, including the cases a = − ∞ or b = ∞. Definition 4.3. The Cumulative Distribution Function (cdf) associated with X (either discrete or continuous) is the. function F (x) = P (X ≤ x). Written out in terms of pdfs and pmfs: F (x) = P (X ≤ x) = ∫ − ∞ x f (t) d t X is continuous ∑ n = − ∞ x p (n) X is discrete By the Fundamental Theorem of Calculus, when X is continuous, F is a continuous function, hence the name continuous rv. The function F is also referred to as the distribution function of X. One major difference between discrete rvs and continuous rvs is that discrete rvs can take on only countably many different values, while continuous rvs typically take on values in an interval such as [0,1] or (− ∞, ∞). Theorem 4.1. Let X be a continuous random variable with pdf f and cdf F. d d x F = f. P (a ≤ X ≤ b) = F (b) − F (a). P (X ≥ a) = 1 − F (a) = ∫ a ∞ f (x) d x. Proof...
eBook - ePubIntroductory Probability and Statistics
Applications for Forestry and Natural Sciences (Revised Edition)
- Robert Kozak, Antal Kozak, Christina Staudhammer, Susan Watts(Authors)
- 2019(Publication Date)
- CAB International(Publisher)
...The cumulative probability function of the exponential density function can be derived from its probability density function and is given as: The cumulative density function is used to calculate probabilities between 0 and a given x value. Example 6.3. Suppose that the time to machine a kitchen cabinet door follows an exponential distribution with a mean of 7 min. a. What is the probability that a door will be completed in less than 4 min? b. What is the probability that a door will be completed in more than 4 min and less than 8 min? a. b. P (4 < X < 8) = P (X < 8)− P (X < 4) ≈ 0.682−0.435 ≈ 0.246. 6.3 Normal Distribution Abraham de Moivre (1667–1754), a French mathematician, derived the mathematical equation of the normal distribution in 1733. Later, Pierre Laplace (1749–1827) and Karl Gauss (1777–1855) further studied and explored the properties of the normal curve. Because of their independent contributions, the normal curve is often called the Gaussian distribution, or in France, the Laplacian distribution. Since its discovery more than 360 years ago, it has become the most important distribution, not only in statistics, but in almost every branch of science. This is because the frequency distributions of many real, natural events follow a normal distribution. It is also because many of the most important theories in statistical inference are based on the normal distribution. We will begin discussing this in detail in Chapter 7. A normal distribution is a symmetric, bell-shaped or mound-shaped distribution (Fig. 6.3), and the general equation of its probability density function is given as: where π = 3.14159 … and e = 2.71828… As shown in Eqn 6.6, the probability function is defined by the mean, μ, and the variance, σ 2 (or standard deviation, σ). The two parameters, μ and σ, respectively specify the position and the shape (or spread) of a normal distribution. To illustrate the effect of the position parameter (μ), Fig...
eBook - ePub- Edward G. Schilling, Dean V. Neubauer(Authors)
- 2017(Publication Date)
- Chapman and Hall/CRC(Publisher)
...3 Probability Functions Many sampling situations can be generalized to the extent that specific functions have proved useful in computing the probabilities associated with the operating characteristic curve and other sampling characteristics. These are functions of a random variable X that take on specific values x at random with a probability evaluated by the function. Such functions are of two types: Frequency function : It gives the relative frequency (or density) for a specific value of the random variable X. It is represented by the function f (x). Distribution function : It gives the cumulative probability of the random variable X up to and including a specific value of the random variable. It can be used to obtain probability over a specified range by appropriate manipulation. It is represented by F (x). In the case of a discrete, go/no-go, random variable f (x) = P (X = x) and the distribution function is simply the sum of the values of the frequency function up to and including x : F (x) = ∑ i = 0 X f (x) X discrete When X is continuous, that is, a measurement variable, it is the integral from the lowest possible value of X, taken here to be –∞, up to x : F (x) = ∫ − ∞ x f (t) d t X continuous where the. notation ∫ a b f (t) d t may be thought of as representing the cumulative probability of f (t) from a lower limit of a to an upper limit of b. In either case, these functions provide a tool for assessment of sampling plans and usually have been sufficiently well tabulated to avoid extensive mathematical calculation. The probability functions can be simply illustrated by a single toss of a six-sided die. Here, the random variable X is discrete and represents the number of spots showing on the upward face of the die. It takes on the values 1, 2, 3, 4, 5, and 6. This is called the sample space...
