One of the most widely known distributions is the Normal Distribution also known as the Gaussian distribution. It is a two-parameter distribution with mean and standard deviation and is symmetrical about its mean. As σ, standard deviation, decreases, the PDF gets squeezed toward the mean. Log-normal Distribution. If the natural logarithm of a given random variable is normally distributed, insteand of the variable then the random variable is said to be log-normally distributed.
The extent to which a distribution of values deviates from symmetry around the mean is the skewness. A value of zero means the distribution is symmetric, while a positive skewness indicates a greater number of smaller values, and a negative value indicates a greater number of larger values (Grad pad, 2013). Values for acceptability for psychometric purposes (+/-1 to +/-2) are the same as with kurtosis. Scatter plots are similar to line graphs in that they both use horizontal and vertical axes to plot data points. The closer the data aims to making a straight line, the higher the correlation between the two variables, or the stronger the relationship(MSTE,n.d) The scatter plot above does not have a straight line formation, so that showing that there is not a strong relationship between the two variables of GPA and final.
The Gaussian distribution—a function that tells the probability that any real observation will fall between any two real limits or real numbers, as the curve approaches zero on either side. It is a very commonly occurring continuous probability distribution. In theory, Gaussian distributions are extremely important in statistics and are often used in the natural and social sciences for real-valued random variables whose distributions are not known. Gaussian distributions are also sometimes referred as Bell curve or normal distribution. The probability density function of a normal (or Gaussian) distribution is defined as Where the parameters μ is the mean or expectation of the distribution σ is the Standard deviation of the distribution σ 2 is the variance of the distribution.
Hence it is also known as normal curve of distribution. Definition of normal distribution: A continuous random variable X is said to follows a normal distribution with parameter μ and σ (or μ and σ2) if the probability functions is f(x) = (1/ σ 2π) e – ½ ((x – μ)/σ)2 ; −∞ < x < ∞, − ∞ < μ < ∞, and σ > 0. This is the formula which is used to show normal curve of distribution. Constants of Normal distribution: Mean = μ Variance = σ2 Standard deviation = σ The graph of the normal curve is shown above. The shape of the curve is bell.
This combination makes the series appear as this : 1+(12)+(12)+(12)+(12)… The series above is divergent because the halfs add up infinitely ther... ... middle of paper ... ...uared ie.x=11n2= 1+14+19+116… This series is convergent using the p-series test because the value of p=2 and when p>1, the series is convergent. In this portfolio, I have investigated information about the harmonic series and then some variations of harmonic series’. To summarize, I have concluded that the harmonic series in divergent through comparing it to another known divergent series and through the improper integral test. I have proved divergence for a few other invented series through the same tests. I also showed the alternating harmonic series and proved that it converged to ln2 using the taylor series.
The most commonly used measures of spread are range, variance and standard deviation. The scales of measurement appropriate for the use of variance and standard deviation are ratio and interval scales. Measures of spread increase on greater variation on the variable. Measures of spread equal zero when there is no variation. Maximum spread for numeric and ordinal variables
Said E.Al-Khamy, defines the fractal dimension D as the measure of the complexity or the space filling ability of the fractal shape. The fractal dimension (a positive real number) is either equal or greater than the topological dimension (a positive integer number). Various formulation and methods exists to find the fractal dimension of fractal (self-similar) structure. One such method which is most popular is formulated as- , Where, D is fractal dimension, N number of parts contained in a self-similar object and, r is the ratio of
These conditions are that a periodic signal must be a piecewise continuous function, have finite number of extrema (maxima and/or minima) and have finite number of discontinuities (points in the function’s domain where the function is not continuous i.e. has a break). Before going to prove the Convergence Theorem, some definitions of terms needs to be clarified. Periodic Function – A function f(x) is periodic with period T if for all the values of x, f(x+nT)=f(x), where T is a positive constant and n is an integer. For example, the sine function sinx is periodic with the least period 2π and other periods such as -2π,4π,6π,
So f(0)=ao. Next, we see that the graph of f1(x)= a0 + a1x will also pass through x= 0, and will have the same slope as f(x) if we let a0=f1(0). Now, if we want to get a better polynomial approximation for this function, which we do of course, we must make a few generalizations. First, we let the polynomial fn(x)= a0 + a1x + a2x2 + ... + anxn approximate f(x) near x=0, and let this functions first n derivatives match the the derivatives of f(x) at x=0. So if we want to make the derivatives of fn(x) equal to f(x) at x=0, we have to chose the coefficients a0 through an properly.
Introduction: In mathematics, the exponential function is the function ex, where e is the number (approximately 2.718281828) such that the function ex equals its own derivative. The exponential function is used to model phenomena when a constant change in the independent variable gives the same proportional change (increase or decrease) in the dependent variable. The exponential function is often written as exp(x), especially when the input is an expression too complex to be written as an exponent. (Source: From Wikipedia) The inverse exponential function is in the form of 1/ ex . Logarithmic functions are inverse of exponential functions.