A function is a mathematical rule that assigns a unique real number to each element of a given set of outcomes. Historically, in the world of statistics and probability, a real function defined over a sample space is traditionally called a random variable.
If the sample space S has a generic element e - the outcome of an experiment, then the random variable is the function X = X(e). Usually, the e is omitted from the notation, so the random variable is denoted by the capital letter X.
A constant is a trivial random variable that always takes the same value, regardless of the outcome of the experiment.
The way in which the probabilities of events involving the random variable X are calculated depends on the nature of that random variable, i.e. from the possibility of counting the set of its possible values.
A random variable is said to be discrete or continuous if the set of its possible values is finite or countably infinite.
A random variable is continuous if it can take on any value from the set of real numbers.
P{X = x} is a probability function and is calculated by summing the probabilities assigned to the desired outcomes. The cumulative distribution function is P{X <= x}. Knowing either of the two functions is sufficient to calculate the probabilities of all events included in P.
The probability function of a continuous variable is called the probability density function.
The distribution of a random variable has two main parameters:
- expected value i
- variance.
Expected value of a random variable
The expected value of a random variable is the mean of the distribution f.
The expected value of a random variable X is calculated as the sum of the products of each possible value of the random variable Xi and the probability of Xi occurring:
The basic property of the expected value is linearity: If X and Y are random variables, and a and b are constants, then
Variance and standard deviation of a random variable variables
The standard deviation is a measure of the dispersion of probabilities around the expected value of a random variable.
The variance of a random variable is calculated as the sum of the squares of the deviations of individual values of the random variable from its expected value.
The standard deviation of a random variable is obtained by calculating the square root of the variance.
Along with the expected value, the variance, or standard deviation, is a parameter of the distribution and is used to describe it.
The applications of the standard deviation of a random variable are numerous. For example, in finance, the standard deviation of a security's price is a measure of the risk carried by that security - the smaller the standard deviation, the lower the risk and vice versa.
expected value
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