How does the variance of $X_1 + X_2$ compare to the variance of $2X$?

The variance of $2X$ is $4\sigma^2$, while the variance of $X_1 + X_2$ is $2\sigma^2$ (assuming independence). Doubling a single observation scales the deviation by 2 and the variance by $2^2$, whereas adding two independent observations allows for some cancellation of errors, resulting in less total variability.

What is the difference between a discrete uniform distribution and a general discrete distribution?

In a discrete uniform distribution, every possible outcome has the exact same probability (e.g., a fair die). In a general discrete distribution, different outcomes can have different probabilities as long as they sum to 1.

When calculating $P(a < X \le b)$ using a cumulative distribution function, what is the correct formula?

The formula is $P(X \le b) - P(X \le a)$. This subtracts the probability of all values up to $a$ from the probability of all values up to $b$, leaving only the values in the interval $(a, b]$.

What is a common mistake when calculating the standard deviation of $Y = 5 - 3X$?

A common error is including the constant '5' or using a negative sign for the spread. The constant '5' shifts the mean but not the SD, and the SD must be positive, so $\sigma_Y = |-3|\sigma_X = 3\sigma_X$.

Why can you not simply add standard deviations when combining two independent random variables?

Standard deviations do not add linearly because they are square roots. You must add the variances (the squared deviations) and then take the square root of the total sum to find the new standard deviation.

What happens if you calculate the variance of $X - Y$ and subtract the variances?

This is an error; variances are always added when combining independent variables, even if the variables themselves are being subtracted. This is because both $X$ and $-Y$ contribute to the total uncertainty/variability of the result.

Define a Discrete Random Variable.

A variable whose possible values form a countable set, meaning you can list the outcomes (even if the list goes on forever). Examples include counts of people, number of events, or money amounts.

What is the 'Expected Value' of a random variable?

The expected value is the mean of the probability distribution. It represents the weighted average of all possible outcomes, where the weights are the probabilities of those outcomes occurring.

What is a Cumulative Probability Distribution?

It is a function or table that expresses the probability that a random variable $X$ will take a value less than or equal to a specific value $x$, denoted as $F(x) = P(X \le x)$.

Why must the sum of all probabilities in a discrete distribution equal 1?

Because the distribution must account for every possible outcome in the sample space. Since one of the outcomes must occur, the total probability of the entire sample space is defined as 1 (or 100%).

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Statistics

Unit 4: Probability, Random Variables & Probability Distributions

Probability Distributions for Discrete Random Variables

Summary

A discrete probability distribution provides a complete description of the probabilities associated with every possible outcome of a discrete random variable. It serves as a mathematical model for random phenomena where outcomes are countable, allowing for the calculation of long-run averages (mean) and variability (standard deviation).

1. Definition & Core Concepts

A Random Variable (RV) is a numerical description of the outcome of a statistical experiment. We typically denote the random variable itself with an uppercase letter (e.g., $X$ ) and its specific possible values with lowercase letters (e.g., $x$ ).
A Discrete Random Variable is one that has a countable number of possible values. This set of values can be finite (like the result of a die roll) or countably infinite (like the number of attempts until a success occurs).
The Probability Distribution of a discrete random variable $X$ lists all possible values $x_i$ and their associated probabilities $P(X = x_i)$ . This can be represented as a table, a probability mass function (PMF), or a histogram.

A bar chart representing a discrete probability distribution where the height of each bar corresponds to the probability of a specific outcome, and the total sum of all bar heights equals 1.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

Probability Distributions for Discrete Random Variables

Q: When calculating $P(a < X \le b)$ using a cumulative distribution function, what is the correct formula?

The formula is $P(X \le b) - P(X \le a)$. This subtracts the probability of all values up to $a$ from the probability of all values up to $b$, leaving only the values in the interval $(a, b]$.

Q: What is a common mistake when calculating the standard deviation of $Y = 5 - 3X$?