What is the fundamental requirement for the sum of probabilities in a discrete distribution?

The sum of all probabilities must equal exactly 1. This ensures that the distribution accounts for every possible outcome in the sample space.

How does $P(X < 3)$ differ from $P(X \le 3)$ for a discrete random variable taking integer values?

$P(X < 3)$ excludes the value 3 (summing probabilities for 0, 1, and 2), whereas $P(X \le 3)$ includes the probability of 3. In discrete sets, the inclusion of the endpoint significantly changes the total.

What is the difference between $E(X^2)$ and $[E(X)]^2$?

$E(X^2)$ is the expected value of the squared outcomes (mean of squares), while $[E(X)]^2$ is the square of the mean. The difference between these two values defines the variance of the distribution.

When calculating $P(X > k)$, how can the complement rule be used?

It can be calculated as $1 - P(X \le k)$. This is particularly useful when the variable has many possible outcomes, making it faster to subtract the unwanted outcomes from the total probability of 1.

What error is likely if a student calculates a negative value for $Var(X)$?

The student likely swapped the terms in the formula or forgot to square the mean. Since variance measures squared deviations, it is mathematically impossible for it to be negative.

Why is it a mistake to assume $E(X)$ must be one of the possible values of $X$?

$E(X)$ is a weighted average, not a mode. For example, the expected value of a fair six-sided die is 3.5, even though a die can never land on 3.5.

What is a common mistake when interpreting the phrase 'at least 2' in a discrete context?

Students often mistakenly use $P(X > 2)$ instead of $P(X \ge 2)$. 'At least' implies that the value itself is the minimum acceptable outcome and must be included in the sum.

Define the Probability Mass Function (PMF).

The PMF is a function that provides the probability $P(X=x)$ for every possible discrete value $x$ that the random variable can take.

What is the formula for the Variance of $X$?

$Var(X) = E(X^2) - [E(X)]^2$. It represents the expected value of the squared deviation of $X$ from its mean.

How do you calculate $E(X)$ for a discrete random variable?

Multiply each possible value $x$ by its corresponding probability $P(X=x)$ and sum all these products together: $\sum x P(X=x)$.

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Probability And Statistics 1

Statistical Distributions

Discrete Probability Distributions

Summary

A discrete probability distribution provides a complete description of a discrete random variable by mapping every possible outcome to its specific probability. It serves as a mathematical model for random events where outcomes are distinct and countable, governed by the fundamental law that the sum of all possible probabilities must equal exactly one.

1. Definition & Core Concepts

Discrete Random Variable (DRV): A variable whose value is determined by the outcome of a random process and can only take specific, distinct values within a set. These values are typically integers or counts, such as the number of successes in a trial.
Probability Mass Function (PMF): A function, often denoted as $P(X = x)$ , that assigns a probability to each possible value of a discrete random variable. For any value $x$ , the probability must satisfy $0 \le P(X = x) \le 1$ .
Representation Methods: Distributions can be expressed as a probability table, where values and probabilities are paired; a formula/function; or a vertical line graph where the height of each line represents the probability of a specific outcome.

A vertical line graph representing a discrete probability distribution where the x-axis shows discrete outcomes and the y-axis shows their respective probabilities.

2. Underlying Principles

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips