7A Calculo De Probabilidades - JulTob/Mathematics GitHub Wiki

🃏 Probability & Statistics

Uncertainty

Random experiment: Any process or action that leads to different possible outcomes, such as rolling a die or flipping a coin. The key feature is that the result cannot be predicted with certainty.
Sample space (Espacio muestral): The set of all possible outcomes. For example, for a coin toss, the sample space is {Heads, Tails}.

Probability function: A function that assigns a probability to each event (a subset of the sample space), which must satisfy:
- $( P(E) \geq 0 )$ for any event $( E )$.
- $( P(\Omega) = 1 )$, where $( \Omega )$ is the entire sample space.
- Additivity: For mutually exclusive events $( E_1, E_2, \ldots )$, $( P(E_1 \cup E_2 \cup \dots) = P(E_1) + P(E_2) + \dots )$.

Classical Probability: If all outcomes in the sample space are equally likely, the probability of an event $( E )$ is:

     P(E) = \frac{\text{Number of favorable outcomes for } E}{\text{Total number of possible outcomes}}

Frequentist Probability: Interprets probability as the long-run relative frequency of an event occurring after many trials.
Subjective Probability: Based on personal belief about how likely an event is.

     P(A \cup B) = P(A) + P(B) - P(A \cap B)

Involves adding more complex structures to the basic probability model to deal with multiple experiments, combinations of events, or more elaborate outcome spaces.

The probability of an event $( A )$, given that another event $( B )$ has occurred, is calculated as:

     P(A | B) = \frac{P(A \cap B)}{P(B)}, \quad \text{provided that } P(B) > 0

Events $( A )$ and $( B )$ are independent if the occurrence of $( A )$ does not affect the probability of $( B )$ (and vice versa):

     P(A \cap B) = P(A) \cdot P(B)

A random variable $( X )$ assigns a numerical value to each outcome of a random experiment.
- Discrete random variables: Take on a finite or countable number of values (e.g., rolling a die).
- Probability mass function (PMF): Describes the probability that a discrete random variable takes a specific value, $( P(X = x) )$.
- Cumulative distribution function (CDF): $( F(x) = P(X \leq x) )$.

The expected value $( E(X) )$ of a random variable $( X )$ is the weighted average of all possible values:

     E(X) = \sum_{x} x \cdot P(X = x)

     \text{Var}(X) = E((X - E(X))^2)

Standard deviation: The square root of the variance, showing how much the values deviate from the mean.
Common discrete distributions include the Binomial and Poisson distributions.

Involves models where experiments are repeated under identical conditions.
For example, the Bernoulli trial models repeated independent trials with two outcomes (success or failure).
The Binomial distribution models the number of successes in a fixed number of independent Bernoulli trials.

Explores the variability inherent in random processes.
Law of Large Numbers: As the number of trials increases, the average of the results converges to the expected value.
Central Limit Theorem: For a large number of trials, the distribution of the sum (or average) of the outcomes tends to follow a normal distribution, even if the original variables are not normally distributed.

Random Experiment: Rolling a fair six-sided die.
Sample Space: $( \Omega = {1, 2, 3, 4, 5, 6} )$.
Probability Assignment: $( P(\text{rolling a 3}) = \frac{1}{6} )$.
Inclusion-Exclusion: $( P(\text{even or } 3) = P({2, 4, 6}) + P({3}) - P(\emptyset) = \frac{3}{6} + \frac{1}{6} = \frac{4}{6} )$.
Conditional Probability: $( P(\text{roll a 3 | roll odd}) = \frac{P({3})}{P({1, 3, 5})} = \frac{\frac{1}{6}}{\frac{3}{6}} = \frac{1}{3} )$.
Random Variable: Let $( X )$ be the result of the roll. $( X )$ can take values 1, 2, 3, 4, 5, or 6.
Expected Value: $( E(X) = \sum_{i=1}^{6} i \cdot P(X = i) = \frac{1+2+3+4+5+6}{6} = 3.5 )$.

That’s a whirlwind summary! If you need more depth on any specific topic, feel free to ask! 🎲📊