* $P(X=0) = 0.25$

* $P(X=1) = 0.50$

* $P(X=2) = 0.25$

The Probability Mass Function (PMF), denoted as $f(x)$ or $P(X=x)$, is a function that gives the probability that a discrete random variable $X$ is exactly equal to some value $x$. It is the specific formula or rule that assigns probabilities to each possible outcome of a discrete random variable.

1. Non-negativity: The probability for any given value $x$ must be non-negative.

$f(x) \ge 0$ for all values of $x$.

2. Summation to One: The sum of all probabilities for all possible values of $x$ must equal 1.

$\sum f(x) = 1$

$f(0) = 0.25$

$f(1) = 0.50$

$f(2) = 0.25$

This satisfies $f(x) \ge 0$ and $0.25 + 0.50 + 0.25 = 1$.

The expected value or mean of a discrete random variable $X$, denoted as $E(X)$ or $\mu$, is the weighted average of all possible values of $X$, where the weights are the probabilities of those values. It represents the long-run average value of the random variable if the experiment were repeated many times.

$E(X) = \sum x \cdot f(x)$

where $x$ represents each possible value of the random variable and $f(x)$ is its corresponding probability.

$E(X) = (0 \cdot 0.25) + (1 \cdot 0.50) + (2 \cdot 0.25) = 0 + 0.50 + 0.50 = 1.0$

The variance of a discrete random variable $X$, denoted as $Var(X)$ or $\sigma^2$, measures the spread or dispersion of the probability distribution around its expected value. A higher variance indicates that the values of the random variable are more spread out from the mean, while a lower variance indicates they are clustered closer to the mean.

$Var(X) = E[(X - E(X))^2] = \sum (x - \mu)^2 f(x)$

where $\mu = E(X)$.

$Var(X) = E(X^2) - [E(X)]^2 = \sum x^2 f(x) - \left(\sum x f(x)\right)^2$

or simply $Var(X) = \sum x^2 f(x) - \mu^2$.

$Var(X) = (0-1)^2 \cdot 0.25 + (1-1)^2 \cdot 0.50 + (2-1)^2 \cdot 0.25$

$Var(X) = (-1)^2 \cdot 0.25 + (0)^2 \cdot 0.50 + (1)^2 \cdot 0.25$

$Var(X) = 1 \cdot 0.25 + 0 \cdot 0.50 + 1 \cdot 0.25 = 0.25 + 0 + 0.25 = 0.50$

First, calculate $E(X^2) = \sum x^2 f(x)$:

$E(X^2) = (0^2 \cdot 0.25) + (1^2 \cdot 0.50) + (2^2 \cdot 0.25)$

$E(X^2) = (0 \cdot 0.25) + (1 \cdot 0.50) + (4 \cdot 0.25) = 0 + 0.50 + 1.00 = 1.50$

Then, $Var(X) = E(X^2) - [E(X)]^2 = 1.50 - (1.0)^2 = 1.50 - 1.00 = 0.50$.

The standard deviation of a discrete random variable $X$, denoted as $\sigma$, is the square root of the variance. It is a more interpretable measure of spread than variance because it is expressed in the same units as the random variable itself.

$\sigma = \sqrt{Var(X)}$

$\sigma = \sqrt{0.50} \approx 0.707$

* Probability Mass Function (PMF): A function, denoted $f(x)$ or $P(X=x)$, that gives the probability that a discrete random variable $X$ takes on a specific value $x$. It must satisfy $f(x) \ge 0$ for all $x$ and $\sum f(x) = 1$.

* Expected Value (Mean): Denoted $E(X)$ or $\mu$, it is the long-run average value of a random variable, calculated as the sum of each possible value multiplied by its probability: $E(X) = \sum x \cdot f(x)$. It represents the center of the distribution.

* Variance: Denoted $Var(X)$ or $\sigma^2$, it is a measure of the spread or dispersion of the values of a random variable around its expected value. It is calculated as $Var(X) = \sum (x - \mu)^2 f(x)$ or $Var(X) = \sum x^2 f(x) - \mu^2$.

* Standard Deviation: Denoted $\sigma$, it is the square root of the variance. It provides a measure of spread in the same units as the random variable, making it easier to interpret than variance. $\sigma = \sqrt{Var(X)}$.

* $P(X=0)$ (TT) = 1/4 = 0.25

* $P(X=1)$ (TH, HT) = 2/4 = 0.50

* $P(X=2)$ (HH) = 1/4 = 0.25

* All $f(x) \ge 0$.

* $\sum f(x) = 0.25 + 0.50 + 0.25 = 1$.

* Expected Value: $E(X) = (0 \cdot 0.25) + (1 \cdot 0.50) + (2 \cdot 0.25) = 1.0$.

* Variance: $Var(X) = (0-1)^2(0.25) + (1-1)^2(0.50) + (2-1)^2(0.25) = 0.25 + 0 + 0.25 = 0.50$.

* Standard Deviation: $\sigma = \sqrt{0.50} \approx 0.707$.

* $P(X=0) = \frac{\binom{3}{0}\binom{7}{3}}{\binom{10}{3}} = \frac{1 \cdot 35}{120} = \frac{35}{120}$

* $P(X=1) = \frac{\binom{3}{1}\binom{7}{2}}{\binom{10}{3}} = \frac{3 \cdot 21}{120} = \frac{63}{120}$

* $P(X=2) = \frac{\binom{3}{2}\binom{7}{1}}{\binom{10}{3}} = \frac{3 \cdot 7}{120} = \frac{21}{120}$

* $P(X=3) = \frac{\binom{3}{3}\binom{7}{0}}{\binom{10}{3}} = \frac{1 \cdot 1}{120} = \frac{1}{120}$

* PMF Verification: Sum of probabilities is $(35+63+21+1)/120 = 120/120 = 1$. All probabilities are non-negative.

* Scenario: Given a function $f(x) = \frac{x}{10}$ for $x = 1, 2, 3, 4$.

* For $x=1, f(1)=0.1$; $x=2, f(2)=0.2$; $x=3, f(3)=0.3$; $x=4, f(4)=0.4$. All are $\ge 0$.

* $\sum f(x) = 0.1 + 0.2 + 0.3 + 0.4 = 1.0$.

* Expected Value: $E(X) = (1 \cdot 0.1) + (2 \cdot 0.2) + (3 \cdot 0.3) + (4 \cdot 0.4) = 0.1 + 0.4 + 0.9 + 1.6 = 3.0$.

* $E(X^2) = (1^2 \cdot 0.1) + (2^2 \cdot 0.2) + (3^2 \cdot 0.3) + (4^2 \cdot 0.4) = (1 \cdot 0.1) + (4 \cdot 0.2) + (9 \cdot 0.3) + (16 \cdot 0.4) = 0.1 + 0.8 + 2.7 + 6.4 = 10.0$.

* $Var(X) = E(X^2) - [E(X)]^2 = 10.0 - (3.0)^2 = 10.0 - 9.0 = 1.0$.

* Standard Deviation: $\sigma = \sqrt{1.0} = 1.0$.

The core of the document revolves around the Probability Mass Function (PMF), denoted as $f(x)$ or $P(X=x)$. The PMF is defined as a function that assigns a probability to each possible value of a discrete random variable $X$. A function qualifies as a PMF only if it satisfies two essential properties: first, all probabilities must be non-negative ($f(x) \ge 0$ for all $x$); and second, the sum of all probabilities for all possible values of $x$ must equal one ($\sum f(x) = 1$). The document illustrates how to construct and verify a PMF using practical examples, such as the number of heads in two coin tosses or the number of defective items in a sample. These examples clearly demonstrate how to list all possible outcomes, assign probabilities, and confirm the PMF properties.

Beyond merely defining the distribution, the document extends to explaining how to characterize it using key statistical measures: expected value (mean), variance, and standard deviation. The expected value, $E(X)$ or $\mu$, represents the long-run average of the random variable. It is calculated as the sum of each possible value multiplied by its corresponding probability: $E(X) = \sum x \cdot f(x)$. This measure provides insight into the central tendency of the distribution.

To quantify the spread or dispersion of the distribution around its mean, the document introduces variance, $Var(X)$ or $\sigma^2$. Variance is defined as the expected value of the squared difference between the random variable and its mean. Two formulas are provided for calculating variance: the direct formula, $Var(X) = \sum (x - \mu)^2 f(x)$, and a more computationally efficient shortcut formula, $Var(X) = \sum x^2 f(x) - \mu^2$. The document emphasizes that variance is expressed in squared units, which can sometimes make it less intuitive.

To address this, the standard deviation, $\sigma$, is introduced as the square root of the variance ($\sigma = \sqrt{Var(X)}$). Standard deviation is a more interpretable measure of spread because it is expressed in the same units as the random variable itself, providing a typical distance of values from the mean.