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".

Explanation

• A Type-1 error, also known as an alpha error.
• It occurs when a statistical test rejects a null hypothesis that is actually true.
• This is a "false positive" result, where a significant effect is detected when none exists.

Explanation

The harmonic mean is the appropriate measure of central tendency to use when calculating the average speed of a journey.

This is because speed is a rate (distance per unit time), and the harmonic mean is used to average rates.

The formula for harmonic mean is: H = n / (∑(1/x))

where H is the harmonic mean, n is the number of values, and x is each value.

Explanation

• The variance of a Chi-square distribution with k degrees of freedom is 2k.
• This means that as the degrees of freedom (k) increase, the variance of the Chi-square distribution also increases.
• It indicating greater spread or dispersion in the data.