College Grade Math College Grade Math Curve Fitting :- Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smooth" function is constructed that approximately fits the data. A related topic is regression analysis, which focuses more on questions of statistical inference such as how much uncertainty is present in a curve that is fit to data observed with random errors. Fitted curves can be used as an aid for data visualization, to infer values of a function where no data are available, and to summarize the relationships among two or more variables. Extrapolation refers to the use of a fitted curve beyond the range of the observed data, and is subject to a greater degree of uncertainty since it may reflect the method used to construct the curve as much as it reflects the observed data. Markov Chains :- A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another,
Know More About :- Axis of Symmetry Formula
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between a finite or countable number of possible states. It is a random process usually characterized as memoryless: the next state depends only on the current state and not on the sequence of events that preceded it. This specific kind of "memorylessness" is called the Markov property. Markov chains have many applications as statistical models of real-world processes. Exponential Growth and Decay :- Are some of the real world problems and it includes all those problems which increases or decreases exponential. Generally the exponential growth of any mathematical function occurs when the growth rate of the value of that mathematical function is proportional to the current value of the function. In some of the special cases it is also called as the geometric growth.The formula for the exponential growth for a variable A at the rate of growth “r” for a specific time interval “t” in the Integer form will be given as: At = A0 (1 + r) t Here At is the value at time t and A0 is the value of the function at initial time t = 0. The examples for the exponential growth can be taken as: The increase in the population of any country, decay of any steel particle due to some reasons, function of exponential as y = 2(5/2)t. So when the growth or decay of any function occurs in some of the exponential form then such type of the growth is called as the exponential growth. Similarly if there is some form of decay in the function then such decay is termed as the exponential decay. The formula for the exponential can be given as: At = A0 (1 - r) t Here also At is the value of function at some time t and A0 is the value of the function at initial time as t = 0. Such a model of exponential growth is also termed as Malthusian growth model.
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Limit Theorems :- Iin Probability are useful for solving the complex Probability and Statistics problems. Here we will discuss some important limit theorems used in probability and statistics: There are following limit theorems in probability and statistics Theorem 1: If event E(Xi Xj) = EXi E Xj, where ‘I’ is not equals to ‘j’, then set of random variables Xi with Exi2 < ∞ are not correlated. Theorem 2: If P(∩j=1n ) Xj = πj=1n Xj, Then set of random variables are independent. Theorem 3: If Xi and Xj are independent, ‘I’ is not equals to ‘j’. Then the set of random variables Xj is pair wise independent. Theorem 4: If S = X1 + X2 +........... +Xn, Then the set of random variables, X1, X2,...........Xn are not correlated. Theorem 5: If a set of random variables X1, X2,...........Xn are sequence of integrable and independent Random Variable, then 1 / n . ∑m=1 n (Xm – E Xm) - > 0. Theorem 6: If a set of random variables X1, X2,...........Xn, are taking values from Sample Space ‘S’, then Xn = X.
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