Probability Chapter: Moment Generating Functions

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Questions and Answers

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?

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?

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?

EX = E(E(X | Y)) Signup and view all the answers

Which of the following statements regarding variance and covariance is true?

Variance is always non-negative, while covariance can be negative. Signup and view all the answers

What is the upper limit of the correlation between two variables X and Y?

1 Signup and view all the answers

What would be the relationship between two random variables if Cov(X, Y) > 0?

X and Y move together positively. Signup and view all the answers

If random variables X and Y are independent, what can be said about Var(X + Y)?

Var(X + Y) = Var(X) + Var(Y) Signup and view all the answers

What is the expected value of the sum of independent and identically distributed random variables X1, X2, ..., Xn?

$n E(X)$ Signup and view all the answers

In the context of independent and identically distributed (iid) random variables, what does 'iid' stand for?

Independently and identically distributed Signup and view all the answers

What is the variance of the sample mean of iid variables with variance $ ext{Var}(X) = au^2$ and sample size n?

$rac{ au^2}{n}$ Signup and view all the answers

What does the joint density function of two random variables X and Y represent?

The likelihood of X and Y occurring together Signup and view all the answers

What is the form of the moment generating function (mgf) for the sum of n iid random variables?

$M_{S_n}(t) = [M_X(t)]^n$ Signup and view all the answers

Which of the following definitions correctly describes the expected value of a function of random variables X and Y?

$E(g(X, Y)) = ext{Integrate } g(x, y) f(x, y) dx dy$ Signup and view all the answers

What is the notation used to denote the joint density function of two random variables?

$f(x, y)$ Signup and view all the answers

What is the variance of the sum $S_n = X_1 + ... + X_n$ of n independent random variables each having variance $ au^2$?

$n au^2$ Signup and view all the answers

What is the primary significance of the central limit theorem?

It states that the distribution of the sample mean approaches a normal distribution as sample size increases. Signup and view all the answers

For a sample size of n, when is the sample mean X approximately normally distributed?

When n is equal to or greater than 30. Signup and view all the answers

What happens if the variables X1, ..., Xn are normally distributed?

The result holds for any sample size n. Signup and view all the answers

How does the delta method relate to the central limit theorem?

It extends the CLT for functions of random variables. Signup and view all the answers

If X is a discrete random variable, what might be necessary for the sample mean to be approximately normally distributed?

A sample size of at least 100. Signup and view all the answers

In the equation X → N(µ, σ²/n), what do the terms represent?

X is the sample mean, µ is the population mean, and σ² is the population variance. Signup and view all the answers

What is a requirement for using the central limit theorem effectively?

Samples must be independent and identically distributed. Signup and view all the answers

For which of the following scenarios is the sample mean X distributed as N(µ, σ²/n)?

When observing outcomes from a binomial distribution with large n. Signup and view all the answers

What does the function MX(t) represent?

The moment-generating function of a random variable X Signup and view all the answers

What is the relationship between the variance and the moment-generating function?

VarX = E(X^2) - (E(X))^2 Signup and view all the answers

If Sn = X1 + · · · + Xn, what is the expected value of Sn?

E(Sn) = E(X1) + ... + E(Xn) Signup and view all the answers

What is the correct formula for the variance of the sum of independent random variables?

Var(Sn) = Sum(Var(Xi)) Signup and view all the answers

What happens when you differentiate MX(t) 'k' times and set t = 0?

You obtain the k-th moment of X Signup and view all the answers

Using the properties of moment-generating functions, how is the moment-generating function of a linear transformation of X expressed?

MaX+b(t) = e^{bt} * MX(at) Signup and view all the answers

What does the notation Var(aX + b) signify in this context?

It represents the effect of scaling a random variable on its variance Signup and view all the answers

What expression represents the moment-generating function for flipping a coin twice where X is the number of heads?

MX(t) = (1/4)e^0 + (1/4)e^t + (1/4)e^{2t} Signup and view all the answers

What does the central limit theorem explain about the normal distribution?

It shows that many variables can have normal distributions. Signup and view all the answers

Which formula accurately represents the Taylor expansion of g(X) around the point µ?

g(X) ≈ g(µ) + g′(µ)(X − µ) Signup and view all the answers

What is the outcome of applying the linear transformation to a normal variable?

It remains a normal distribution with shifted mean and variance. Signup and view all the answers

In the formula g(X) = g′(µ) X + g(µ) − g′(µ)µ, what does g(µ) represent?

The value of the function g(X) evaluated at mean µ. Signup and view all the answers

What distribution does the sum X1 + · · · + Xn approach as n becomes large?

Normal distribution Signup and view all the answers

If EX = µ and VarX = σ2, what are the mean and variance of the transformed variable g(X)?

Mean: g(µ), Variance: [g′(µ)]^2σ2 Signup and view all the answers

In the context of the central limit theorem, what does the notation N(nµ, nσ²) signify?

The distribution of the sum of n independent variables. Signup and view all the answers

What does the term 'Bernoulli Trials' represent in a statistical context?

Independent trials with two possible outcomes. Signup and view all the answers

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.