5. Discrete Random Variables I

This paragraph introduces the concept of random variables in the context of probability theory. It explains that random variables assign numerical values to outcomes of experiments, making the subject more interesting and complex. The paragraph discusses how to define and describe random variables using probability mass functions (PMFs), which assign probabilities to different numerical outcomes. The speaker also mentions the importance of understanding the new concept of expected value and briefly touches on the concept of variance, setting the stage for the topics that will be covered in the unit.

The speaker delves deeper into the concept of random variables, emphasizing their nature as functions mapping from the sample space to real numbers. Using the example of measuring the height of a student, the paragraph illustrates how numerical values are assigned to each outcome. It distinguishes between the random variable as an abstract function (denoted by a capital letter) and the specific numerical values it can take (denoted by lowercase letters). The paragraph also introduces the idea that multiple random variables can be derived from a single experiment, such as height and weight of a student, and discusses how functions of random variables are also random variables themselves.

This paragraph further explores the concept of random variables, discussing their types, which can be discrete or continuous. The speaker uses the example of measuring height in inches versus a more precise scale to illustrate discrete and continuous random variables, respectively. The importance of understanding discrete random variables before moving on to continuous ones is highlighted, as the latter introduces complexities of calculus. The paragraph also emphasizes the distinction between the random variable itself and the numerical values it takes, using the notation of capital letters for variables and lowercase for specific values.

The speaker introduces PMFs as a way to describe the likelihood of different numerical values that a random variable can take. PMFs are represented as a bar graph where the height of each bar corresponds to the probability of the random variable taking on a particular value. The paragraph explains how to calculate PMFs by summing the probabilities of all outcomes that lead to a specific numerical value. It also provides an example of a geometric PMF, which describes the probability of the number of coin tosses until the first head appears, and discusses the properties of PMFs, such as all probabilities being non-negative and their sum equaling one.

The paragraph provides examples of calculating PMFs for discrete random variables, using the scenarios of coin tossing and dice rolling. It explains how to determine the probability of a random variable taking a specific value by considering all possible outcomes that result in that value. The speaker illustrates this with a detailed example of rolling a tetrahedral die twice and calculating the PMF for the minimum value obtained. The concept of a geometric random variable is also revisited, with an emphasis on the calculation of PMFs for such variables.

The speaker discusses the Central Limit Theorem, a fundamental concept in probability theory, which states that the distribution of the sum (or average) of a large number of independent and identically distributed variables approaches a normal distribution, regardless of the original distribution. This is illustrated by plotting the PMF of the number of heads obtained in a series of coin tosses and observing how it begins to resemble a bell curve as the number of tosses increases. The paragraph also emphasizes the importance of understanding the PMFs of discrete random variables before tackling more complex continuous random variables.

This paragraph introduces the expected value of a random variable, which is a central concept in probability theory. The expected value is described as an average of the possible outcomes, weighted by their probabilities. The speaker provides an example using a game with different rewards and probabilities to illustrate the calculation of expected value. The paragraph also offers an alternative interpretation of expected value as the center of gravity of the distribution, which can be used to quickly calculate it in some cases, such as symmetric distributions.

The speaker explains how to calculate the expected value of a function of a random variable, using the law of the unconscious statistician. This law allows for the calculation of the expected value of a function by considering the original random variable's PMF, rather than finding the PMF of the function itself. The paragraph provides a mathematical justification for this law and highlights the importance of understanding when it is appropriate to apply this shortcut in calculations. It also includes a cautionary note that the average of a function of a random variable is not generally the same as the function of the average.

This paragraph discusses the properties of expectations, particularly their linearity. It explains that the expected value of a linear function of a random variable is equal to the linear function of the expected value. The speaker provides examples, such as multiplying a random variable by a constant or adding a constant to a random variable, and demonstrates how these operations affect the expected value. The paragraph also emphasizes that while expectations are generally averages, the function of an average is not the same as the average of the function, except in the case of linear functions.

The speaker introduces variance as a measure of how spread out the distribution of a random variable is. Variance is defined as the expected value of the squared deviation from the mean and gives more weight to larger deviations due to the squaring operation. The paragraph explains how to calculate variance using a practical formula that involves the expected value of the random variable and the expected value of its square. It also discusses the properties of variance, such as its non-negativity and how it changes under linear transformations, specifically that multiplying a random variable by a constant increases the variance by the square of that constant.

In the final paragraph, the speaker concludes the discussion on variance and its behavior under linear transformations. It reiterates that multiplying a random variable by a constant affects the variance by increasing it by the square of that constant, while adding a constant to a random variable does not change its variance. The paragraph wraps up the session and informs the audience that the next meeting will be on Wednesday.