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What does convergence in law state about random variables?
What does convergence in law state about random variables?
The Strong Law of Large Numbers (SLLN) states that the sample mean converges to the expected value almost surely.
The Strong Law of Large Numbers (SLLN) states that the sample mean converges to the expected value almost surely.
True
What is the limit distribution of Zn according to the Central Limit Theorem?
What is the limit distribution of Zn according to the Central Limit Theorem?
N(0, 1)
The Poisson theorem states that binomial random variables converge in law to the Poisson distribution if npn approaches ______ as n approaches infinity.
The Poisson theorem states that binomial random variables converge in law to the Poisson distribution if npn approaches ______ as n approaches infinity.
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Match the following terms with their correct descriptions:
Match the following terms with their correct descriptions:
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What type of convergence does the Central Limit Theorem provide?
What type of convergence does the Central Limit Theorem provide?
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What is the Berry–Esseen theorem about?
What is the Berry–Esseen theorem about?
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Convergence of characteristic functions is equivalent to convergence in law.
Convergence of characteristic functions is equivalent to convergence in law.
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What does the Weak Law of Large Numbers (WLLN) state about the sample means?
What does the Weak Law of Large Numbers (WLLN) state about the sample means?
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The Central Limit Theorem applies only to samples that are normally distributed.
The Central Limit Theorem applies only to samples that are normally distributed.
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What is the use of the indicator variable in opinion polls?
What is the use of the indicator variable in opinion polls?
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The Strong Law of Large Numbers (SLLN) states that the sample means converge _____ to the actual population mean as n approaches infinity.
The Strong Law of Large Numbers (SLLN) states that the sample means converge _____ to the actual population mean as n approaches infinity.
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Match the following terms with their definitions:
Match the following terms with their definitions:
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In the context of the Central Limit Theorem, what happens to the distribution of sample means as sample size increases?
In the context of the Central Limit Theorem, what happens to the distribution of sample means as sample size increases?
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Convergence in probability requires that the probability of the difference between Xn and Y is strictly positive for infinite n.
Convergence in probability requires that the probability of the difference between Xn and Y is strictly positive for infinite n.
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Explain what is meant by convergence almost everywhere.
Explain what is meant by convergence almost everywhere.
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What does the characteristic function uniquely determine?
What does the characteristic function uniquely determine?
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The Strong Law of Large Numbers (SLLN) applies only to independent random variables.
The Strong Law of Large Numbers (SLLN) applies only to independent random variables.
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What is the moment generating function of a random variable X?
What is the moment generating function of a random variable X?
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The exponent of the characteristic function is of the form $e^{itX}$, where ___ represents a constant.
The exponent of the characteristic function is of the form $e^{itX}$, where ___ represents a constant.
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Match the following concepts with their definitions:
Match the following concepts with their definitions:
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For which type of distribution is the characteristic function $ rac{ ext{λ}}{ ext{λ} - it}$ defined?
For which type of distribution is the characteristic function $ rac{ ext{λ}}{ ext{λ} - it}$ defined?
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The Central Limit Theorem states that the sum of a large number of independent and identically distributed random variables is normally distributed.
The Central Limit Theorem states that the sum of a large number of independent and identically distributed random variables is normally distributed.
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What is the main difference between Weak Law of Large Numbers (WLLN) and Strong Law of Large Numbers (SLLN)?
What is the main difference between Weak Law of Large Numbers (WLLN) and Strong Law of Large Numbers (SLLN)?
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Study Notes
Probability Theory and Statistics Lecture Notes
- Rostyslav Hryniv, Autumn 2018
- Ukrainian Catholic University - Computer Science and Business Analytics Programs
- 3rd term
Law of Large Numbers and Central Limit Theorem
-
Inequalities and Weak Law of Large Numbers
- Motivation for the law
- Relevant inequalities
- Description of the weak law
-
Convergence and Strong Law of Large Numbers
- Convergence in probability
- Convergence almost everywhere
- Description of the strong law
-
Central Limit Theorem
- Characteristic functions
- Convergence in law
- Convergence of sums
Frequentist Approach to Probability
-
Definition
- Frequentist probability defines an event's probability
- Relative frequency in a large number of trials
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Justification by Law of Large Numbers
- Problems with definition
- Determining what constitutes "large enough"
- Potential variations when repeated
Sums of Independent Identically Distributed Random Variables
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Independent Identically Distributed (i.i.d.) Random Variables
- Defining i.i.d. random variables
- Expectations (E(X₁)) and Variations (Var(X₁))
-
Sums and Sample Mean (Mn)
- Defining the sum (Sn)
- Variance of Sₙ
- Sample mean definition (Mn)
- Expectation of sample mean (E(Mn))
- Variance of sample mean (Var(Mn))
-
Law of Large Numbers (informal)
- Describing convergence to the true mean (μ)
Markov Inequality
-
Definition
- Defining a non-negative random variable (r.v. X)
- Defining a constant (a > 0)
- Defining the inequality: P(X ≥ a) ≤ E(X)/a
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Proof
- Auxiliary r.v. (Ya) definition
- Showing that Ya ≤ X
- Using expectation to prove the inequality
-
Example
- Application of Markov Inequality
Chebyshev's Inequality
-
Definition
- Defining a random variable (X) with mean (μ) and variance (σ²)
- Defining a constant (c > 0)
- Defining the inequality: P(|X – μ| ≥ c) ≤ σ²/c²
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Proof
- Showing that {|X – μ| ≥ c} = {(X – μ)² ≥ c²}
- Using the Markov inequality
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Example
- Application of Chebyshev's Inequality
One-Sided Chebyshev's Inequality
-
Definition
- Defining the one-sided inequality:
P(X - μ ≥ c) ≤ σ²/(σ² + c²)
- Defining the one-sided inequality:
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Proof
- Using the definition of the event to derive the inequality
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Example
- Illustrative example using application of the one-sided Chebyshev's Inequality
Derivation of the Weak Law of Large Numbers (WLLN)
-
Introduction of i.i.d. random variables and sample means
- Identifying independent random variables (X₁, X₂,...)
- Defining the sample mean (Mn) and its expectation (E(Mn)) and variance(Var(Mn))
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Applying Chebyshev's Inequality
- Relation between sample mean and Chebyshev's inequality
- Expressing the probability in terms of variance, sample size and constant
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Result
- Probability converges to 0 as sample size increases
- Result: the process eventually converges to the true mean (μ)
- Probability converges to 0 as sample size increases
Weak Law of Large Numbers (WLLN)
-
Theorem
-
Independent and identically distributed random variables
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Defining the mean (μ)
- Defining probability of difference: P(|Mn - μ| ≥ ε) = 0 as n approaches infinity
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Example
- Describing the example with normal distribution (N (μ, σ²) and sample mean
Example: Opinion Poll
-
Introduction
- Sampling from a population
- Assessing support for a topic (T)
- Relationship between support rate (p) and sample observations
-
Estimation of Support Rate
- Using k/n as an estimate (p)
- defining I and indicator (Ij) the Bernoulli indicator
-
Discussion of Large Sample Size
- Determining needed sample size (n) to achieve certain probability levels and desired precision
Convergence in Probability
-
Definition
- Sequence of r.v.'s (X₁, X₂,...) converging in probability to r.v.Y
- Using the probability expression for the convergence
-
WLLN Implication
- The sample means (Sn) of i.i.d. random variables eventually converge to the underlying mean.
Convergence Almost Everywhere
-
Definition
- Defining convergence almost everywhere (a.e).
- Expressing almost sure convergence (Xn → X)
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Example
- Illustrative example with uniform distribution
-
Theorem
- Implication of convergence a.e. to convergence in probability
Strong Law of Large Numbers (SLLN)
-
Theorem (SLLN)
- Independent and identically random variables convergence
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Corollary (estimating cdf)
- Defining the indicator random (variables) Ik
- Convergence of empirical probability distribution
Empirical c.d.f.
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Definition
- Defining the empirical c.d.f.
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Theorem (Glivenko-Cantelli)
Monte Carlo
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SLLN application in Monte Carlo simulation
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Examples
- Illustrative example involving continuous function and uniformly distributed r.v.'s
Characteristic Functions
- Defining characteristic function:
Φx(t) = E(eitx) - Relation between characteristic functions and probability density functions
- Uniqueness and moments of distribution related to characteristic functions
Examples
-
Normal Distribution
- Defining normal distribution and expression of characteristic function
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Exponential Distribution
Convergence in Law
-
Definition
- Define convergence in law
-
Example (Poisson Approximation)
- Illustrative example including convergence in law for binomial and Poisson
Central Limit Theorem (CLT)
-
Assumptions for CLT
- Independent and identically distributed random values
- Definition of the sample mean
- Sample mean converges to underlying mean
-
Standardized sums (Zn)
-
Result of CLT
Berry-Esseen Theorem
- Rate of convergence in CLT
Opinion Polls
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Task (Opinion Polls) -Estimating fraction(p) that supports a topic(T)
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Indicators (Ik)
-
Sample Means and CLT -Convergence of estimated fractions
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Example for n -Determining necessary sample size for a specific confidence level and interval
Example: Post Office Overweight
- Context -Determining the probability that a total weight is below a given value
- Random Variables (Xn)
- Calculations -Using sum of random variables and Chebyshev's Inequality.
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Description
Explore the fundamental concepts of Probability Theory and Statistics, focusing on the Law of Large Numbers and the Central Limit Theorem. This quiz covers key topics such as Frequentist probability, convergence types, and relevant inequalities. Test your understanding and application of these essential statistical principles.
