Combinations of Random Variables

• Understanding combinations of random variables is fundamental. These involve the combination of two or more random variables in different styles such as addition, subtraction, multiplication, or division.

• In context of statistics, a random variable could be a numerical outcome of a random occurrence. Variables could be continuous or discrete. Discrete random variables take distinct or separate values whereas continuous variables take an infinite number of possible values within an interval.

• The sum or difference of two or more independent random variables also creates a random variable. Its expected value, or mean, is the sum or difference of the individual means.

• The variance of the sum or difference of two or more independent random variables is the sum of the individual variances, regardless of whether the variables are being added or subtracted.

• If two variables X and Y are not independent, then the covariance must be taken into account. Covariance measures the extent to which the relationship between the two variables is linear. If covariance is positive, then as one variable increases the other also tends to increase. If the covariance is negative then as one variable increases the other tends to decrease.

• The variance of the sum of two dependent random variables is the sum of their variances plus twice their covariance.

• When multiplying a random variable by a scalar, the expectation gets multiplied by the scalar, but the variance gets multiplied by the square of the scalar.

• Moments of random variables in terms of combination introduces skewness and kurtosis of the distribution. Skewness measures the lack of symmetry, while kurtosis measures the “tailedness”.

• The Central Limit Theorem states that the sum of a large number of independent and identically-distributed random variables, each with finite mean and variance, will approximately follow a normal distribution.

• The concepts of correlation and regression come into play when analysing relationships between two variables. Correlation measures the strength and direction of a linear relationship between two variables, while regression is a statistical method for investigating functional relationships between variables.

• Bivariate Normal Distribution allows to characterise the joint distribution of two different normal random variables including the correlation between them.

• The primary application of combinations of random variables is in the use of statistical estimators, which are functions of the data (and therefore random variables themselves) that give estimates of parameters under the scope.

Keep referring to sample problems and theorems in order to ensure that you understand the concepts clearly. Practice exercises regularly to reiterate your understanding and to prepare for any question types that might come. Unlocking the understanding of combinations of random variables gives a better grip over data prediction, research, and the overall subject of statistics.