Key Concepts in Number Theory and Statistics
Number Theory and Modular Arithmetic
1. Define GCD (1 Mark)
GCD (Greatest Common Divisor) is the largest positive integer that divides two or more integers exactly.
2. Prime Factorization of 1330 (1 Mark)
1330 = 2 × 5 × 7 × 19
3. Define Residue (1 Mark)
A residue modulo m is any integer equivalent to another integer modulo m.
4. State the Chinese Remainder Theorem (1 Mark)
If the moduli are pairwise coprime, a system of congruences has a unique solution modulo the product of the moduli.
5. Define Fermat Numbers (1 Mark)
A Fermat number is a number of the form: Fn = 22n + 1, where n ≥ 0.
6. Define Congruences (1 Mark)
Two integers a and b are congruent modulo m if: a ≡ b (mod m) ⇔ m | (a – b).
Statistical Measures and Probability
1. Define Correlation (1 Mark)
Correlation is the statistical measure that indicates the degree and direction of the relationship between two variables.
2. Define Regression (1 Mark)
Regression is a statistical method used to estimate or predict the value of one variable from another variable.
4. Define Random Variable (1 Mark)
A random variable is a function that assigns a real number to each outcome of a random experiment.
5. What is Expectation? (1 Mark)
Expectation (Expected Value) is the mean or average value of a random variable.
6. Define Binomial Distribution (1 Mark)
A Binomial distribution is the probability distribution of the number of successes in n independent Bernoulli trials with success probability p.
7. Define Poisson Distribution (1 Mark)
A Poisson distribution is a discrete probability distribution that gives the probability of a given number of events occurring in a fixed interval.
10. State Bernoulli’s Distribution (1 Mark)
A Bernoulli distribution is a discrete distribution with two possible outcomes: success (probability p) and failure (probability q = 1 – p; also noted as q = 1 – p, q = 1 – p).
Binomial Distribution Mean Calculation
8. Mean of Binomial Distribution (1 Mark)
Find the mean if n = 3 and p = 9/4 (also noted as p = 94 and p = 4/9 in source).
Mean: μ = n × p = 3 × (9/4) = 27/4 = 6.75 (Calculation sequence: 3 × 9/4 = 27/4 = 6; 3 × 9/4 = 27/4 = 6.75).
Final Mean: 6.75
Continuous Distributions and Sampling
1. Define Normal Distribution (1 Mark)
A Normal Distribution is a continuous probability distribution that is bell-shaped and symmetric about its mean.
2. Define t-Distribution (1 Mark)
The t-distribution is a continuous probability distribution used for small samples when the population standard deviation is unknown.
3. Define F-Distribution (1 Mark)
The F-distribution is a continuous probability distribution used to compare two population variances.
4. State the Central Limit Theorem (1 Mark)
The Central Limit Theorem states that for a large sample size, the sampling distribution of the sample mean is approximately normal, regardless of the population distribution.
5. Define Sampling Distribution (1 Mark)
A sampling distribution is the probability distribution of a statistic (such as the sample mean) obtained from all possible samples of a fixed size.
Statistical Inference and Estimation
1. Define Population with Example (1 Mark)
A population is the entire collection of individuals or objects under study. Example: All students in a college.
2. Define Estimate (1 Mark)
An estimate is a numerical value calculated from sample data to approximate an unknown population parameter.
3. Define Point Estimation (1 Mark)
Point estimation is the process of using a single value from a sample to estimate a population parameter.
4. Formula for Single Mean (Large Samples) (1 Mark)
The Z-test statistic for a single mean is: Z = (X̄ – μ) / (σ / √n), where X̄ = sample mean, μ = population mean, σ = population standard deviation, and n = sample size.
5. Formula for Two Means (Large Samples) (1 Mark)
The Z-test statistic for two means is: Z = (X̄₁ – X̄₂) / √((σ₁²/n₁) + (σ²²/n²)).
6. Define Chi-square Test (1 Mark)
The Chi-square (χ²) test is a statistical test used to compare observed and expected frequencies to determine whether there is a significant difference. (Note: Mean = 6.75 uiwhd).
Stochastic Processes and Markov Chains
1. Define Stochastic Process (1 Mark)
A stochastic process is a collection of random variables indexed by time or space, representing the evolution of a random phenomenon.
2. Define Markov Chain (1 Mark)
A Markov chain is a stochastic process in which the future state depends only on the present state and not on the past states.
3. Define Aperiodic (1 Mark)
A state in a Markov chain is aperiodic if it can be revisited at irregular time intervals, i.e., its period is 1.
4. Define Irreducible (1 Mark)
A Markov chain is irreducible if every state can be reached from every other state in a finite number of steps.
5. State the Transition Probability Matrix (1 Mark)
A transition probability matrix is a square matrix whose element Pij represents the probability of moving from state i to state j, and each row sums to 1.