Statistics: Relation to Normal Distribution

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Questions and Answers

What is required for the convolution formula to compute the distribution of the sum of random variables?

The random variables must be identically distributed.

The random variables must be independent.

The random variables can be either independent or continuous.

The random variables must be continuous. (correct)

Which statement is correct regarding uniform distributions under linear transformations?

They are uniformly distributed only if the transformation is one-to-one.

They are uniformly distributed in all regions.

They will not remain uniformly distributed under any circumstances.

They remain uniformly distributed only in certain regions. (correct)

What does the density function of the sum Z = X1 + X2 represent?

It is independent of the individual densities.

It only applies to iid random variables.

It is computed as the convolution of their densities. (correct)

It can also represent the product of two densities.

How is the density of the product Z = X1 * X2 calculated?

By integrating the product of their densities over a specific range. Signup and view all the answers

For the joint pdf of (Y1, Y2) defined as Y1 = X1 + X2 and Y2 = X1 - X2, which of the following is true?

It exists for all continuous random variables. Signup and view all the answers

What does the ratio of independent random variables Z = X1 / X2 represent?

It has a density that depends on both marginal densities. Signup and view all the answers

Which of the following scenarios allows for the convolution of two probability density functions?

If both are continuous and independent distributions. Signup and view all the answers

What is the characteristic of the pdf for the difference Z = X1 - X2 of independent random variables?

It can be negative or positive, depending on the variables. Signup and view all the answers

What is the relationship between a Student t random variable and the standard normal and chi-square distributions?

A Student t random variable is the ratio of a standard normal variable to a chi-square variable. Signup and view all the answers

What value range can an F random variable take?

(0, ∞) Signup and view all the answers

Which of the following statements correctly describes the transformation of a discrete random variable X into Y = T(X)?

The pmf of Y is given by pY(yj) = P(T(X) = yj) = Σ pX(xi) for all xi that satisfy T(xi) ≤ yj. Signup and view all the answers

How can one compute the probability density function (pdf) of Y = T(X) if the cdf is impractical to calculate?

By computing the pdf of Y = T(X) from the underlying distributions. Signup and view all the answers

In the context of chi-square distributions, if X1 and X2 are independent chi-square variables with n1 and n2 degrees of freedom, what is the distribution of Z = X1/n1 / X2/n2?

Z follows an F distribution with n1 and n2 degrees of freedom. Signup and view all the answers

What method can be used to find the distribution of the sum of two independent standard exponential random variables X and Y?

The distribution of Z is the convolution of the distributions of X and Y. Signup and view all the answers

What is the requirement for a random variable Y that is defined as a transformation T of another random variable X?

The distribution of X should be known and well defined. Signup and view all the answers

Which type of distribution does the ratio formed by two independent chi-square random variables with their respective degrees of freedom primarily represent?

F distribution Signup and view all the answers

Under what conditions can the hypergeometric distribution with parameters N, M, n be directly approximated by the normal distribution?

N, M, n are all large, and n is much larger than N, M Signup and view all the answers

Which of the following statements is true regarding chi-square random variables?

The sum of independent chi-square variables is another chi-square variable. Signup and view all the answers

How is the chi-square distribution related to the normal distribution?

It is derived from the sum of independent standard normal random variables squared. Signup and view all the answers

True or false: some exponential random variable is a chi-square random variable.

True Signup and view all the answers

What is the relationship between the chi-square distribution and the Gamma distribution?

Chi-square is a special case of the Gamma distribution. Signup and view all the answers

What characterizes the Student's t random variable?

It has a density function that is defined for all real numbers. Signup and view all the answers

In which context is the Gamma distribution used?

To describe the distribution of the sum of independent chi-square variables. Signup and view all the answers

What is the primary use of the Student's t distribution?

To estimate population parameters when population variance is unknown. Signup and view all the answers

Study Notes

Relation to the Normal Distribution

The Student t random variable with n degrees of freedom can be defined using a standard normal random variable (X) and a chi-square random variable (Y) with n degrees of freedom, where X and Y are independent.
The F random variable with n1 and n2 degrees of freedom is related to the normal distribution through the chi-square distribution.
An F random variable with n1 and n2 degrees of freedom can be defined using two independent chi-square random variables (X1 and X2) with n1 and n2 degrees of freedom respectively.

Transformations of Random Variables

To find the distribution of a random variable Y that is a function (T) of another random variable X with a known distribution, we need to determine the distribution of Y.
We can compute the cumulative distribution function (cdf) of Y by considering the probability that Y is less than or equal to a given value y.
For discrete random variables, the cdf of Y can be computed by summing the probabilities of all values of X for which T(X) is less than or equal to y.
For continuous random variables, the cdf of Y can be computed by integrating the probability density function (pdf) of X over all values of x for which T(x) is less than or equal to y.
If computing the cdf of Y is impractical, we can try computing the pmf or pdf of Y directly.

Approximations

The Central Limit Theorem states that the distribution of the sample mean of independent and identically distributed random variables approaches a normal distribution as the sample size increases.
The normal distribution can be used to approximate other distributions, such as the binomial, Poisson, and hypergeometric distributions, under certain conditions.
The hypergeometric distribution can be directly approximated by the normal distribution if the parameters N, M, and n are all large and n is much smaller than N and M.

The chi-square distribution with n degrees of freedom is a continuous distribution that takes values in the interval (0,∞).
The chi-square distribution can be defined as the sum of squares of n independent standard normal random variables.
The chi-square distribution is a special case of the gamma distribution.
The Student t distribution with n degrees of freedom is another continuous distribution that can take any real value.
The Student t distribution is related to the normal distribution and the chi-square distribution.

Standard Transformations

The convolution formula can be used to compute the distribution of the sum of two independent continuous random variables.
The convolution formula involves integrating the product of the densities of the two random variables over all possible values of one of the variables.
The density of the difference of two independent continuous random variables can be computed using a similar formula.
The density of the product of two independent continuous random variables can also be computed using a formula that involves integrating the product of the densities of the two variables.
The density of the ratio of two independent continuous random variables can be computed using a formula that involves integrating the product of the densities of the two variables.

Conditional Distributions

The joint distribution of two random variables describes the probability of observing specific values for both variables.
The marginal distribution of a random variable describes the probability of observing specific values for that variable, regardless of the values of the other variables.
The conditional distribution of one random variable given the value of another random variable describes the probability of observing specific values for the first variable, given that a specific value has been observed for the second variable.

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Description

This quiz covers key concepts regarding the Student t and F random variables and their relation to the normal distribution. It also addresses transformations of random variables and how to compute their cumulative distribution functions. Test your understanding of these fundamental statistical principles!