Prove that "independent" implies "uncorrelated" and construct an example to show that the converse is not true.

Understanding the concept of independent random variables is crucial to grasping various statistical principles. Independent variables in statistics are two or more variables that do not influence each other. In other words, knowledge about the outcome of one does not provide any information about the outcome of the other.

For example, imagine tossing a coin and rolling a die. The results of the coin toss won't affect what number the die rolls; these two actions are independent of each other. Mathematically, we say that two random variables X and Y are independent if the probability of X occurring alongside Y is equal to the product of their individual probabilities:
\( P(X=x, Y=y) = P(X=x) \times P(Y=y) \).

It is important to emphasize that independent variables can result in all kinds of relationships when studied empirically - they may seem related by chance alone, but under the hood, there is no direct influence one holds over the other.

Another key concept in statistics is the correlation coefficient. It is a measure that denotes the strength and direction of a linear relationship between two variables. The value of the correlation coefficient ranges from -1 to +1.

A value of +1 implies a perfect positive linear relationship, where variables move in the same direction, whereas a value of -1 implies a perfect negative linear relationship, where they move in opposite directions. A value of 0 indicates that there is no linear correlation between the variables.

The formula for calculating the correlation coefficient, denoted as \( Corr(X,Y) \), is:
\( Corr(X,Y) = \frac{Cov(X,Y)}{\sigma_X \sigma_Y} \),

where \( Cov(X,Y) \) is the covariance of X and Y, and \( \sigma_X \) and \( \sigma_Y \) are their standard deviations, respectively. Understanding this relationship is essential for statistic analysis, as it helps to decipher how variables may change in response to one another.

Diving further into statistical relationships, covariance is a measure used to determine how two variables change together. If the greater values of one variable correspond with greater values of another, and similarly for lesser values, the covariance is positive. Conversely, if greater values of one variable mainly correspond with lesser values of another, the covariance is negative.

Mathematically, the covariance between two random variables X and Y is calculated as:
\( Cov(X,Y) = E[(X - E[X])(Y - E[Y])] \).

If X and Y are independent, the covariance will be zero because the variables do not affect each other, and therefore the product of their deviations from their means is also zero in expectation. However, a zero covariance does not guarantee that variables are independent, as they may have a non-linear relationship that it does not capture.

Lastly, let's talk about statistical independence. This term specifically refers to the notion that the occurrence of one event does not affect the probability of the occurrence of another event. In terms of random variables, this means that the presence of a certain value for one variable does not sway the probability distribution of the other variable.

Technically, for two events A and B, we say they are independent if the probability of both occurring equals the product of the probabilities of each occurring separately:
\( P(A \text{ and } B) = P(A) \times P(B) \).

Statistical independence is a foundational concept in probability and is often a necessary assumption in various statistical methods and tests. When two random variables are independent, it simplifies analysis and the interpretation of data, because the effect of one variable does not need to be adjusted for the effects of the other.