Explanation

There are 12 face cards in a standard deck of 52 cards (3 face cards per suit x 4 suits).

The probability of drawing 2 face cards is: C(12,2) x C(40,3) / C(52,5) = 0.0533

Explanation

• A sample statistic is a numerical value that summarizes a characteristic of a sample.
• Such as the sample mean, sample proportion, sample variance, etc.

Explanation

• In the Cramér-Rao inequality, var(T) is the minimum variance bound, which represents the lowest possible variance of an unbiased estimator T.
• This bound is a fundamental limit in estimation theory and is used to evaluate the efficiency of estimators.

Explanation

The Poisson distribution is a discrete probability distribution that models the number of events (k) occurring in a fixed interval of time or space.

The probability mass function of the Poisson distribution is:

P(k) = (e^(-m) * (m^k)) / k!

where m is the mean (expected value) of the distribution, and e is the base of the natural logarithm (approximately 2.718).

If the mean (m) of a Poisson distribution is 1, then the probability of P(1) is:

P(1) = (e^(-1) * (1^1)) / 1! = 1/e ≈ 0.3679

So, the probability of P(1) is approximately 0.368, or 1/e.

Explanation

• When the number of unknowns is greater than the number of equations, the system is underdetermined, and there is no unique solution.
• In such cases, we use the method of least squares to estimate the unknowns.

Explanation

• The variance of a Poisson distribution is equal to its mean, which is denoted by m.
• This means that if a random variable X follows a Poisson distribution with mean m, then the variance of X is also m.

Explanation

The basic principle of experimental design that is NOT listed is: Confounding.

Explanation

• The Rao-Blackwell theorem enables us to obtain minimum variance unbiased estimates through sufficient statistics, which capture all the information in the data about the parameter.
• By conditioning an unbiased estimator on a sufficient statistic, we can improve its precision and obtain the minimum variance unbiased estimator (MVUE).

Explanation

Explanation

• The sum of deviations from the regression line is zero.
• This is a fundamental property of linear regression, known as the "zero-sum property" or "error-sum property".