Is it true that Mutually exclusive events are independent.?

Mutually exclusive events are dependent because if one occurs, the other cannot. Independence means the occurrence of one does not affect the probability of the other.

Is it true that The expected value of a discrete random variable must be one of the possible values the variable can take.?

The expected value represents the average outcome over many trials and can be a value not present in the set of possible outcomes for the random variable.

Probability Rules and Distributions Cheat Sheet

This cheat sheet consolidates key probability rules, including the Multiplicative Rule, Law of Total Probability, and Bayes' Theorem, alongside essential discrete and continuous probability distributions like Binomial, Poisson, Uniform, and Normal, providing formulas and properties for practical application.

Core Principles

Multiplicative Rule: P(A ∩ B) = P(A|B)P(B) or P(A ∩ B) = P(B|A)P(A) to find the probability of two events occurring.
Law of Total Probability: P(B) = Σ P(B|Aᵢ)P(Aᵢ) allows calculation of event B's probability by conditioning on a set of mutually exclusive and exhaustive events Aᵢ.
Bayes' Theorem: P(Aᵢ|B) = [P(B|Aᵢ)P(Aᵢ)] / Σ P(B|Aⱼ)P(Aⱼ) updates the probability of an event Aᵢ given new evidence B.
Independence: Two events A and B are independent if P(A|B) = P(A), meaning the occurrence of one does not affect the probability of the other.
Probability Mass Function (pmf) Properties: For a discrete RV, p(x) ≥ 0 for all x, and the sum of p(x) over all possible x must equal 1.
Probability Density Function (pdf) Properties: For a continuous RV, f(x) ≥ 0 for all x, and the integral of f(x) over all real numbers must equal 1.

Action Steps

Apply the Multiplicative Rule: Use P(A ∩ B) = P(A|B)P(B) or P(A ∩ B) = P(B|A)P(A) to calculate the joint probability of two events.
Utilize the Law of Total Probability: Employ P(B) = Σ P(B|Aᵢ)P(Aᵢ) to find the probability of an event B by considering all possible scenarios defined by mutually exclusive and exhaustive events Aᵢ.
Apply Bayes' Theorem: Update prior probabilities of events Aᵢ to posterior probabilities P(Aᵢ|B) using the formula when new information B is available.
Check for Independence: Verify if P(A|B) = P(A) or P(A ∩ B) = P(A)P(B) to determine if events A and B are independent.
Calculate Expected Value: Use E(X) = Σ x * p(x) for discrete RVs or E(X) = ∫ x * f(x) dx for continuous RVs to find the average outcome.
Calculate Variance: Use V(X) = E[(X - μ)²] or V(X) = E(X²) - [E(X)]² to measure the spread of a distribution.
Identify Distribution Type: Match problem characteristics (e.g., fixed trials, success/failure, rate) to the appropriate distribution (Binomial, Poisson, Exponential, etc.) to apply its specific formulas.

Formulas

Multiplicative Rule: $P(A \cap B) = P(A|B)P(B)$ or $P(A \cap B) = P(B|A)P(A)$
Law of Total Probability: $P(B) = \sum_{i=1}^{k} P(B|A_i)P(A_i)$
Bayes' Theorem: $P(A_j | B) = \frac{P(B|A_j)P(A_j)}{\sum_{i=1}^{k} P(B|A_i)P(A_i)}$
Independence: $P(A \cap B) = P(A) \cdot P(B)$
Probability Mass Function (pmf) Properties: $p(x) \geq 0$ for any x, and $\\sum p(x) = 1$
Cumulative Distribution Function (CDF) for discrete RV: $F(x) = P(X \leq x) = \sum_{y \leq x} p(y)$
Probabilities of Intervals from Discrete CDFs: $P(a < X \leq b) = F(b) - F(a)$
Expected Value (discrete RV): $\\mu_x = E(X) = \sum_{x} x \cdot p(x)$
Expected Value of a Function of X (discrete RV): $E[h(X)] = \sum_{D} h(x) \cdot p(x)$
Variance (discrete RV): $V(X) = \sigma_x^2 = \sum_{D} (x - \mu)^2 \cdot p(x) = E[(X - \mu)^2]$
Standard Deviation (discrete RV): $\\sigma_x = \sqrt{V(X)}$
Variance Proposition 1: $V(X) = E(X^2) - \mu^2$
Variance Proposition 2: $V(a+bX) = b^2V(X)$
Moment-Generating Function (MGF) (discrete RV): $M_X(t) = E(e^{tX}) = \sum_{D} e^{tx} p(x)$
Bernoulli pmf: $bern(x;p) = p^x (1-p)^{1-x}, \quad x = 1 \text{ or } 0$
Bernoulli Expected Value: $E(X) = p$
Bernoulli Variance: $V(X) = p(1-p)$
Binomial pmf: $b(x;n,p) = \binom{n}{x} p^x (1-p)^{n-x}, \quad x = 0, 1, 2, ..., n$
Binomial Expected Value: $E(X) = np$
Binomial Variance: $V(X) = np(1-p)$
Geometric pmf: $g(x; p) = p(1-p)^x, \quad x = 0, 1, 2, ...$
Geometric Expected Value: $E(X) = \frac{1-p}{p}$
Geometric Variance: $V(X) = \frac{1-p}{p^2}$
Hypergeometric pmf: $h(x; n, M, N) = \frac{\binom{M}{x} \binom{N-M}{n-x}}{\binom{N}{n}}$
Hypergeometric Expected Value: $E(X) = n \frac{M}{N}$
Hypergeometric Variance: $V(X) = n \frac{M}{N} (1 - \frac{M}{N}) \frac{N-n}{N-1}$
Poisson pmf: $p(x; \mu) = \frac{e^{-\mu} \mu^x}{x!}, \quad x = 0, 1, 2, ...$
Poisson Expected Value: $E(X) = \mu$
Poisson Variance: $V(X) = \mu$
Continuous RV pdf: $f(x) \geq 0$ for all $x \in R$ and $\\int_{-\\infty}^{\\infty} f(x) dx = 1$
Continuous RV Probability: $P(a \leq X \leq b) = \int_{a}^{b} f(x) dx$
Continuous RV Expected Value: $\\mu_x = E(X) = \int_{-\\infty}^{\\infty} x \cdot f(x) dx$
Continuous RV Expected Value of a Function: $E[h(X)] = \int_{-\\infty}^{\\infty} h(x) \cdot f(x) dx$
Continuous RV Variance: $V(X) = \sigma_x^2 = \int_{-\\infty}^{\\infty} (x - \mu)^2 \cdot f(x) dx = E[(X - \mu)^2]$
Uniform pdf: $f(x; A, B) = \frac{1}{B-A}, \quad A \leq x \leq B$
Uniform Expected Value: $E(X) = \frac{A+B}{2}$
Uniform Variance: $V(X) = \frac{(B-A)^2}{12}$
Normal pdf: $f(x; \mu, \sigma) = \frac{1}{\\sqrt{2 \pi} \sigma} e^{-(x-\mu)^2 / 2 \sigma^2}$
Normal Expected Value: $E(X) = \mu$
Normal Variance: $V(X) = \sigma^2$
Standard Normal pdf: $f(z; 0,1) = \frac{1}{\\sqrt{2 \pi}} e^{-z^2/2}, \quad - \infty < z < \infty$
Standard Normal CDF: $\\(z) = \frac{1}{\\sqrt{2 \pi}} \int_{-\\infty}^{z} e^{-t^2/2} dt$
Transform to Standard Normal: $Z = \frac{X - \mu}{\\sigma}$
Exponential pdf: $f(x; \lambda) = \\lambda e^{-\\lambda x}, \quad x \geq 0$
Exponential Expected Value: $E(X) = 1/\\lambda$
Exponential Variance: $V(X) = 1/\\lambda^2$
Gamma pdf: $f(x; \alpha, \beta) = \frac{x^{\\alpha-1}e^{-x/\beta}}{\\beta^\alpha \Gamma(\\alpha)}, \quad x \geq 0$
Gamma Expected Value: $E(X) = \alpha \beta$
Gamma Variance: $V(X) = \alpha \beta^2$
Gamma Function: $\\Gamma(\\alpha) = \int_{0}^{\\infty} t^{\\alpha-1} e^{-t} dt$
Gamma Function Properties: $\\Gamma(\\alpha+1) = \alpha \\Gamma(\\alpha)$, $\\Gamma(1) = 1$, $\\Gamma(n) = (n-1)!$, $\\Gamma(1/2) = \sqrt{\\pi}$
Chi-Squared pdf: $f(x; v) = \frac{x^{v/2-1}e^{-x/2}}{2^{v/2}\Gamma(v/2)}, \quad x \geq 0$
Chi-Squared Expected Value: $E(X) = v$
Chi-Squared Variance: $V(X) = 2v$
Standard Beta pdf: $f(x; \alpha, \beta) = \frac{\\Gamma(\\alpha + \beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)} \cdot x^{\\alpha-1}(1-x)^{\\beta-1}, \quad 0 \leq x \leq 1$
Standard Beta Expected Value: $E(X) = \frac{\\alpha}{\\alpha + \beta}$
Standard Beta Variance: $V(X) = \frac{\\alpha \\beta}{(\\alpha + \beta)^2 (\\alpha + \beta + 1)}$
Poisson Approximation to Binomial: $b(x; n,p) \rightarrow p(x; \mu)$ when $n \rightarrow \infty$ and $p \rightarrow 0$ such that $np \rightarrow \mu > 0$
Normal Approximation to Binomial: $P(X \leq x) \approx \Phi\left(\frac{x + 0.5 - np}{\\sqrt{np(1-p)}}\right)$ for large $n$ and $p$ where $np > 10$ and $n(1-p) > 10$

Key Terms

Conditional Probability: The probability of an event occurring given that another event has already occurred, denoted P(A|B).
Mutually Exclusive Events: Events that cannot occur at the same time; their intersection is empty.
Exhaustive Events: A set of events that covers all possible outcomes in the sample space.
Prior Probability: The initial probability of an event before considering new evidence.
Posterior Probability: The updated probability of an event after considering new evidence, calculated using Bayes' Theorem.
Probability Mass Function (pmf): A function that gives the probability that a discrete random variable is exactly equal to some value.
Probability Density Function (pdf): A function whose integral over any interval gives the probability that a continuous random variable falls within that interval.
Expected Value: The weighted average of all possible values of a random variable, weighted by their probabilities; also known as the mean.
Variance: A measure of the spread or dispersion of a set of data or a probability distribution around its mean.
Standard Deviation: The square root of the variance, providing a measure of dispersion in the same units as the data.
Moment-Generating Function (MGF): A function used to generate the moments of a probability distribution.
Memoryless Property: A property of the exponential distribution where the probability of a future event is independent of the past.

Pro Tips

Use the 'choose successes × choose failures / choose total' mnemonic for hypergeometric problems.
Recognize that the Gamma(1, β) distribution is identical to the Exponential(1/β) distribution.
Understand that the Chi-Squared distribution is a special case of the Gamma distribution with specific parameters.
When approximating a Binomial distribution with a Normal distribution, ensure np > 10 and n(1-p) > 10 for a reliable approximation.
Remember that adding a constant to a random variable does not change its variance: V(a+X) = V(X).

Pitfalls to Avoid

Confusing mutual exclusivity with independence: Mutually exclusive events cannot occur together, while independent events do not influence each other's probability.
Incorrectly applying formulas: Using discrete formulas for continuous variables or vice-versa can lead to significant errors.
Ignoring conditions for approximations: Using Poisson or Normal approximations without meeting the required conditions (e.g., np > 10) can yield inaccurate results.
Misinterpreting variance units: Variance is in squared units, while standard deviation has the same units as the random variable, making standard deviation more practical for interpretation.

Myth vs Reality

Mutually exclusive events are independent.: Mutually exclusive events are dependent because if one occurs, the other cannot. Independence means the occurrence of one does not affect the probability of the other.
The expected value of a discrete random variable must be one of the possible values the variable can take.: The expected value represents the average outcome over many trials and can be a value not present in the set of possible outcomes for the random variable.

Real World Examples

Calculating the probability of drawing two Aces from a standard deck of cards without replacement.: This scenario uses the Multiplicative Rule: P(Ace on 1st draw) * P(Ace on 2nd draw | Ace on 1st draw).
A medical test for a rare disease has a low false positive rate and a low false negative rate.: Bayes' Theorem is crucial for calculating the actual probability of having the disease given a positive test result, considering the low prevalence of the disease.
Determining the probability of a machine part being defective based on different manufacturing shifts, each with its own defect rate.: The Law of Total Probability is used, where the overall defect probability is the sum of probabilities of defects from each shift, weighted by the proportion of production from each shift.
A quality control process involves sampling items from a batch without replacement.: The Hypergeometric distribution is used to calculate the probability of finding a certain number of defective items in the sample.
Modeling the number of customer arrivals at a store per hour.: The Poisson distribution is suitable for modeling the number of events occurring in a fixed interval of time or space when events are independent and occur at a constant average rate.

Statistics

Empirical Rule - Within 1 Standard Deviation: 68% of data falls within 1 standard deviation of the mean for a normal distribution.
Empirical Rule - Within 2 Standard Deviations: 95% of data falls within 2 standard deviations of the mean for a normal distribution.
Empirical Rule - Within 3 Standard Deviations: 99.7% of data falls within 3 standard deviations of the mean for a normal distribution.