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Questions and Answers
What is the formula used to express covariance?
What is the formula used to express covariance?
- Cov(X, Y) = E(XY) + (EX)(EY)
- Cov(X, Y) = E(X)E(Y)
- Cov(X, Y) = E(X - Y)
- Cov(X, Y) = E(XY) - (EX)(EY) (correct)
Which property of covariance indicates its symmetry?
Which property of covariance indicates its symmetry?
- Cov(X, Y) = Cov(Y, X) (correct)
- Cov(X + Y, U + V) = Cov(X, V)
- Cov(X, X) = 0
- Cov(aX + b, cY + d) = 0
What is the significance of Cov(X, Y) being equal to zero?
What is the significance of Cov(X, Y) being equal to zero?
- X and Y have a strong linear relationship.
- X and Y have a positive correlation.
- Covariance cannot be calculated for X and Y.
- X and Y are independent random variables. (correct)
What does the law of total expectation state?
What does the law of total expectation state?
Which of the following statements regarding variance and covariance is true?
Which of the following statements regarding variance and covariance is true?
What is the upper limit of the correlation between two variables X and Y?
What is the upper limit of the correlation between two variables X and Y?
What would be the relationship between two random variables if Cov(X, Y) > 0?
What would be the relationship between two random variables if Cov(X, Y) > 0?
If random variables X and Y are independent, what can be said about Var(X + Y)?
If random variables X and Y are independent, what can be said about Var(X + Y)?
What is the expected value of the sum of independent and identically distributed random variables X1, X2, ..., Xn?
What is the expected value of the sum of independent and identically distributed random variables X1, X2, ..., Xn?
In the context of independent and identically distributed (iid) random variables, what does 'iid' stand for?
In the context of independent and identically distributed (iid) random variables, what does 'iid' stand for?
What is the variance of the sample mean of iid variables with variance $ ext{Var}(X) = au^2$ and sample size n?
What is the variance of the sample mean of iid variables with variance $ ext{Var}(X) = au^2$ and sample size n?
What does the joint density function of two random variables X and Y represent?
What does the joint density function of two random variables X and Y represent?
What is the form of the moment generating function (mgf) for the sum of n iid random variables?
What is the form of the moment generating function (mgf) for the sum of n iid random variables?
Which of the following definitions correctly describes the expected value of a function of random variables X and Y?
Which of the following definitions correctly describes the expected value of a function of random variables X and Y?
What is the notation used to denote the joint density function of two random variables?
What is the notation used to denote the joint density function of two random variables?
What is the variance of the sum $S_n = X_1 + ... + X_n$ of n independent random variables each having variance $ au^2$?
What is the variance of the sum $S_n = X_1 + ... + X_n$ of n independent random variables each having variance $ au^2$?
What is the primary significance of the central limit theorem?
What is the primary significance of the central limit theorem?
For a sample size of n, when is the sample mean X approximately normally distributed?
For a sample size of n, when is the sample mean X approximately normally distributed?
What happens if the variables X1, ..., Xn are normally distributed?
What happens if the variables X1, ..., Xn are normally distributed?
How does the delta method relate to the central limit theorem?
How does the delta method relate to the central limit theorem?
If X is a discrete random variable, what might be necessary for the sample mean to be approximately normally distributed?
If X is a discrete random variable, what might be necessary for the sample mean to be approximately normally distributed?
In the equation X → N(µ, σ²/n), what do the terms represent?
In the equation X → N(µ, σ²/n), what do the terms represent?
What is a requirement for using the central limit theorem effectively?
What is a requirement for using the central limit theorem effectively?
For which of the following scenarios is the sample mean X distributed as N(µ, σ²/n)?
For which of the following scenarios is the sample mean X distributed as N(µ, σ²/n)?
What does the function MX(t) represent?
What does the function MX(t) represent?
What is the relationship between the variance and the moment-generating function?
What is the relationship between the variance and the moment-generating function?
If Sn = X1 + · · · + Xn, what is the expected value of Sn?
If Sn = X1 + · · · + Xn, what is the expected value of Sn?
What is the correct formula for the variance of the sum of independent random variables?
What is the correct formula for the variance of the sum of independent random variables?
What happens when you differentiate MX(t) 'k' times and set t = 0?
What happens when you differentiate MX(t) 'k' times and set t = 0?
Using the properties of moment-generating functions, how is the moment-generating function of a linear transformation of X expressed?
Using the properties of moment-generating functions, how is the moment-generating function of a linear transformation of X expressed?
What does the notation Var(aX + b) signify in this context?
What does the notation Var(aX + b) signify in this context?
What expression represents the moment-generating function for flipping a coin twice where X is the number of heads?
What expression represents the moment-generating function for flipping a coin twice where X is the number of heads?
What does the central limit theorem explain about the normal distribution?
What does the central limit theorem explain about the normal distribution?
Which formula accurately represents the Taylor expansion of g(X) around the point µ?
Which formula accurately represents the Taylor expansion of g(X) around the point µ?
What is the outcome of applying the linear transformation to a normal variable?
What is the outcome of applying the linear transformation to a normal variable?
In the formula g(X) = g′(µ) X + g(µ) − g′(µ)µ, what does g(µ) represent?
In the formula g(X) = g′(µ) X + g(µ) − g′(µ)µ, what does g(µ) represent?
What distribution does the sum X1 + · · · + Xn approach as n becomes large?
What distribution does the sum X1 + · · · + Xn approach as n becomes large?
If EX = µ and VarX = σ2, what are the mean and variance of the transformed variable g(X)?
If EX = µ and VarX = σ2, what are the mean and variance of the transformed variable g(X)?
In the context of the central limit theorem, what does the notation N(nµ, nσ²) signify?
In the context of the central limit theorem, what does the notation N(nµ, nσ²) signify?
What does the term 'Bernoulli Trials' represent in a statistical context?
What does the term 'Bernoulli Trials' represent in a statistical context?
Study Notes
Moment Generating Function (mgf)
- Defined as MX(t) = Ee^tX
- Useful for determining moments of a random variable
- The kth moment is obtained by differentiating MX(t) k times and setting t = 0
- For a linear function of X (aX + b), the mgf is MaX+b(t) = e^(tb) MX(at)
Sums of Random Variables
- Let Sn = X1 + ... + Xn be the sum of n random variables
- E(Sn) = E(X1) + ... + E(Xn)
- Var(Sn) = Var(X1) + ... + Var(Xn) if the variables are independent
- The mgf of Sn is MSn(t) = MX1(t) * ... * MXn(t)
- If X1, ... , Xn are iid, meaning they have the same distribution and are independent, then:
- E(Sn) = nEX
- Var(Sn) = nVarX
- MSn(t) = [MX(t)]^n
Sample Mean
- The sample mean is defined as X = (X1+...+Xn)/n = Sn/n
- Mean of the sample mean: EX = µ
- Variance of the sample mean: VarX = σ^2/n
Multiple Random Variables
- Joint density function: f(x, y)
- Expected value of g(X, Y): E[g(X, Y)] = ∫∫g(x, y)f(x, y) dx dy
- Covariance: Cov(X, Y) = E[(X - EX)(Y - EY)] = E(XY) - (EX)(EY)
- Correlation: Corr(X, Y) = Cov(X, Y) / (√VarXVarY) ∈ [-1, 1]
Law of Total Expectation
- Also known as the tower law or law of iterated expectation
- States: EX = E[E(X|Y)]
Central Limit Theorem (CLT)
- States that for a large sample of iid variables with mean µ and variance σ^2, the distribution of the sample mean approaches a normal distribution with mean µ and variance σ^2/n
- Typically, n ≥ 30 is enough for the CLT to hold
- If the variables are themselves normally distributed, the CLT holds for any value of n
Delta Method
- Extends the CLT to differentiable functions of random variables
- States that for a differentiable function g(x), g(X) is approximately normally distributed with mean g(µ) and variance [g'(µ)]^2 σ^2/n
Other Useful Results
- Bernoulli Trials:
- The mean of a Bernoulli distribution is p and the variance is p(1-p), where p is the probability of success
- The sample mean of n Bernoulli trials has a normal distribution for large n (n ≥ 30) with mean p and variance p(1-p)/n
- Normal Approximation for Proportions:
- If X is the number of successes in n Bernoulli trials, then (X/n - p) / √(p(1-p)/n) has a standard normal distribution for large n.
- This is used for hypothesis testing and confidence interval construction.
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Description
This quiz focuses on moment generating functions (mgf), sums of random variables, and the sample mean. Explore key concepts such as the definition of mgf, its application in determining moments, and the properties of sums and averages of random variables. Perfect for enhancing your understanding of probability theory.
