# Define the term stochastic models?

Question
Modeling data distributions
Define the term stochastic models?

2020-11-08
Stochastic modeling is a form of financial model that is used to help make investment decisions. This type of modeling forecasts the probability of various outcomes under different conditions, using random variables. Stochastic modeling presents data and predicts outcomes that account for certain levels of unpredictability or randomness. Companies in many industries can employ stochastic modeling to improve their business practices and increase profitability. In the financial services sector, planners, analysts, and portfolio managers use stochastic modeling to manage their assets and liabilities and optimize their portfolios. Deterministic modeling produces constant results Deterministic modeling gives you the same exact results for a particular set of inputs, no matter how many times you re-calculate the model. Here, the mathematical properties are known. None of them is random, and there is only one set of specific values and only one answer or solution to a problem. With a deterministic model, the uncertain factors are external to the model. Stochastic modeling produces changeable results Stochastic modeling, on the other hand, is inherently random, and the uncertain factors are built into the model. The model produces many answers, estimations, and outcomes—like adding variables to a complex math problem—to see their different effects on the solution. The same process is then repeated many times under various scenarios. Stochastic modeling is used in a variety of industries around the world. The insurance industry, for example, relies heavily on stochastic modeling to predict how company balance sheets will look at a given point in the future. Other sectors, industries, and disciplines that depend on stochastic modeling include stock investing, statistics, linguistics, biology, and quantum physics. A stochastic model incorporates random variables to produce many different outcomes under diverse conditions.

### Relevant Questions

Define the term Mean.
In these problem you are asked to find a function that models a real-life situation. Use the principles of modeling described in this Focus to help you. Radius find a function that models the radius r of a circle in term of its area A.

The article “Stochastic Modeling for Pavement Warranty Cost Estimation” (J. of Constr. Engr. and Mgmnt., 2009: 352–359) proposes the following model for the distribution of Y = time to pavement failure. Let $$\displaystyle{X}_{{{1}}}$$ be the time to failure due to rutting, and $$\displaystyle{X}_{{{2}}}$$ be the time to failure due to transverse cracking, these two rvs are assumed independent. Then $$\displaystyle{Y}=\min{\left({X}_{{{1}}},{X}_{{{2}}}\right)}$$. The probability of failure due to either one of these distress modes is assumed to be an increasing function of time t. After making certain distributional assumptions, the following form of the cdf for each mode is obtained: $$\displaystyle\Phi{\left[\frac{{{a}+{b}{t}}}{{\left({c}+{\left.{d}{t}\right.}+{e}{t}^{{{2}}}\right)}^{{\frac{{1}}{{2}}}}}\right]}$$ where $$\Uparrow \Phi$$ is the standard normal cdf. Values of the five parameters a, b, c, d, and e are -25.49, 1.15, 4.45, -1.78, and .171 for cracking and -21.27, .0325, .972, -.00028, and .00022 for rutting. Determine the probability of pavement failure within $$\displaystyle{t}={5}$$ years and also $$\displaystyle{t}={10}$$ years.

This problem is about the equation
dP/dt = kP-H , P(0) = Po,
where k > 0 and H > 0 are constants.
If H = 0, you have dP/dt = kP , which models expontialgrowth. Think of H as a harvesting term, tending to reducethe rate of growth; then there ought to be a value of H big enoughto prevent exponential growth.
Problem: find acondition on H, involving $$\displaystyle{P}_{{0}}$$ and k, that will prevent solutions from growing exponentially.
Two drugs, Abraxane and Taxol, are both cancer treatments, yet have differing rates at which they leave a patient’s system. Using terminology from pharmacology, Abraxane leaves the system by so-called “first-order elimination”, which means that the concentration decreases at a constant percentage rate for each unit of time that passes. Taxol leaves the system by “zero-order elimination”, which means that the concentration decreases by a constant amount for each unit of time that passes.
(a) As soon as the infusion of Taxol is completed, the drug concentration in a patient’s blood is 1000 nanograms per milliliter $$\displaystyle{\left(\frac{{{n}{g}}}{{{m}{l}}}\right)}.$$ 12 hours later there is $$\displaystyle{50}\frac{{{n}{g}}}{{{m}{l}}}$$ left in the patient’s system. Use the data to construct an appropriate formula modeling the blood concentration of Taxol as a function of time after the infusion is completed.
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$$\displaystyle{y}={0.069}{x}-{4.755}.$$
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