What is Random Variable?

A random variable (RV), denoted by an uppercase letter like X, Y, or Z, is a real-valued function defined over the elements of a sample space. Lowercase letters (x, y, z) represent specific values that the random variable can take. An example is provided using a coin-tossing experiment where X represents the number of heads.

For an experiment of tossing a coin twice, the sample space consists of {HH, TH, HT, TT}. If X is defined as the number of heads obtained, then HH maps to 2, TH maps to 1, HT maps to 1, and TT maps to 0. Thus, the specific values for X (small x) can be 0, 1, and 2. The video also explains notation like P(X <= x) for probabilities.

Random variables can be categorized as discrete or continuous. Discrete random variables take on distinct, countable values (e.g., number of cars, number of dots on a dice), often whole numbers. Continuous random variables can take any value within a given range, representing an infinite number of possible values (e.g., height, weight, distance traveled).

An experiment of rolling a pair of dice is used to illustrate a discrete random variable. The possible outcomes are limited to whole numbers. A tree diagram is used to visualize outcomes, leading to a sample space of 36 possible elements, representing the combinations of dots on two dice.

If X is defined as the total number of points obtained when rolling two dice, the video demonstrates how each element in the sample space maps to a specific value of X. For example, (1,1) maps to 2, (1,2) maps to 3, and so on. The possible specific values for X (small x) range from 2 to 12.

Another example involves selecting two socks from a drawer containing five brown and three green socks. A tree diagram is used to determine the sample space, which includes combinations like (Brown, Brown), (Brown, Green), (Green, Brown), and (Green, Green). This example also touches upon the concept of conditional probability, as socks are removed in succession without replacement.

If W is defined as the number of brown socks chosen, the possible values for W are 0, 1, and 2. The probabilities for each value are calculated by multiplying along the branches of the tree diagram. For example, P(W=0) is P(Green, Green), P(W=1) is P(Brown, Green) + P(Green, Brown), and P(W=2) is P(Brown, Brown). The results are presented in a table.

The video concludes by demonstrating how to find the probability of specific events, such as getting both brown socks, which corresponds to P(W=2), calculated as 20/56.