Elements of Probability Relevant to Modeling

In its most simplest form, a random variable is a variable that changes unpredictably or randomly. If some variable is repeatedly measured and every time a new observation is obtained, that variable is random or stochastic. If the same value is obtained every time, the variable is fixed or deterministic. Mathematically, a random variable is a function defined over some sample 

Every possible outcome of a random variable is associated with a probability for that event occurring. Two functions map outcome to probability for continuous random variables the probability density function (pdf) and cumulative distribution function (cdf). In the discrete case, the pdf and cdf are referred to as the probability mass function and cumulative mass function, respectively. A function f(x) is a pdf for some continuous random variable X if and only if 

If X is a random variable with pdf f(x) then the cumulative probability of an event x occurring, denoted F(x), is 

F(x) is called the cdf and it is the integral of the pdf evaluated from —oo to x. For random variables that cannot take on negative values, then the integral is evaluated from 0 to x. Clearly, if F(x) is a cdf then dF(x)/dx is the pdf. The cdf is a monotonically increasing function with limits equal to 0 as x approaches —oo and 1 as x approaches +oo. By changing the limits of 

Knowing the pdf of a distribution is useful because then the expected value and variance of the distribution may be derived. The expected value of a pdf is its mean and can be thought of as the center of mass of the density. If X is a continuous random with domain — oo, oo variable then its expected value is 

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