Discrete Random Variables and Binomial Distribution: Key Concepts and Formulas

Discrete Random Variables

Expected Value and Standard Deviation

Discrete random variables are variables that can take on a countable number of distinct values. Two important characteristics of discrete random variables are their expected value (mean) and standard deviation, which measure the central tendency and spread of their probability distributions.

• Expected Value (Mean): The expected value of a discrete random variable $X$ is given by:

$\mu = \sum_{x \in X} x P(x)$

• Standard Deviation: The standard deviation of $X$ is:

$\sigma = \sqrt{\sum_{x \in X} (x - \mu)^2 P(x)}$

• Interpretation: The expected value represents the long-run average outcome, while the standard deviation quantifies the variability around the mean.

• Example: If $X$ represents the number of heads in three coin tosses, the expected value and standard deviation can be calculated using the probabilities for each possible outcome.

Binomial Distribution

Definition and Properties

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, each with the same probability of success.

• Notation: $X \sim B(n, p)$ means that $X$ follows a binomial distribution with parameters $n$ (number of trials) and $p$ (probability of success).

• Parameters:

  • $n$ = number of independent trials

  • $p$ = probability of success in each trial

  • $q$ = probability of failure in each trial ($q = 1 - p$)

• Possible Values: $X$ can take integer values from $0$to$n$(i.e.,$x = 0, 1, 2, ..., n$).

Key Formulas

• Probability Mass Function: The probability of observing exactly $x$ successes in $n$ trials is:

$P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$

• Mean (Expected Value):

$\mu = np$

• Standard Deviation:

$\sigma = \sqrt{np(1-p)}$

• Example: If a fair coin is tossed 10 times ($n=10$, $p=0.5$), the expected number of heads is $\mu = 10 \times 0.5 = 5$, and the standard deviation is $\sigma = \sqrt{10 \times 0.5 \times 0.5} \approx 1.58$.

Comparison Table: Binomial Distribution Parameters

Parameter	Symbol	Description
Number of trials	n	Total number of independent experiments
Probability of success	p	Chance of success in a single trial
Probability of failure	q	$q = 1 - p$
Mean	$\mu$	$\mu = np$
Standard deviation	$\sigma$	$\sigma = \sqrt{np(1-p)}$

Geometric Distribution

Introduction

The geometric distribution is another discrete probability distribution, which models the number of trials needed to achieve the first success in repeated, independent Bernoulli trials.

• Key Properties:

  • Each trial is independent.

  • Probability of success ($p$) is constant for each trial.

  • Random variable $X$ represents the trial number of the first success.

• Example: If the probability of rolling a six on a die is $p = 1/6$, the geometric distribution models the number of rolls needed to get the first six.

Additional info: The geometric distribution is briefly mentioned in the notes, but not elaborated. Standard formulas for mean and variance are:

$\text{Mean}: \mu = \frac{1}{p}$ $\text{Variance}: \sigma^2 = \frac{1-p}{p^2}$