Final answer:
A random variable represents numerical outcomes of a probability experiment and can be discrete or continuous. A Bernoulli random variable denotes the outcome of an experiment with two possible outcomes, generally coded as 1 for success and 0 for failure. It is a special case of a binomial distribution with one trial.
Step-by-step explanation:
A random variable is a variable whose values are numerical outcomes of a probability experiment. These values can change each time the experiment is conducted. Random variables can be either discrete or continuous. Discrete random variables have countable values, often resulting from counting occurrences, such as the number of heads in coin tosses. Continuous random variables, on the other hand, have uncountable values, typically obtained from measurements, like the height of students.
A Bernoulli random variable represents the outcome of a Bernoulli trial, which is an experiment that has exactly two possible outcomes: success or failure. This type of random variable is a special case of a binomial distribution with only one trial (n=1). It usually takes the value of 1 for a success and 0 for a failure, and thus can be used to model binary events, such as a coin landing heads (success) or tails (failure).
Uppercase letters like X or Y are used to denote a random variable, whereas lowercase letters like x or y represent the value of a random variable. For example, if X represents the number of successes in n trials of a Bernoulli experiment, the values of X could be the integers from 0 to n, and the probability distribution function (PDF) would show the probability of each possible value of X.