Elementary Probability and Statistics Midterm 2

A point estimate of a parameter is a single number that can be regarded as a sensible value for the parameter

The cumulative distribution F(x) = P(X>x)

The median of a continuous distribution, denoted by μ is the 50th percentile

If X and Y are independent, then p = 0, but, p = 0 does not imply independence

A parameter is any quantity whose value can be calculated from sample data

The beta distribution provides positive density only for X in an interval of finite length

A nonnormal distribution is said to be symmetric and light tailed if the density curve declines more rapidly out in the

A nonnegative random variable X is said to have a lognormal distribution if the random variable Y = ln(X) has normal dis

The first population moment is (1/n) * sigmaX

When choosing among several different estimators of theta, select one that is biased

The use of the word marginal is the term marginal probability mass function is a consequence of the fact that if the joi

A continuous distribution whose pdf is symmetric has median \overline{μ} equal to the point of symmetry

For f(x) to be a legitamate probab density function, the are under the curve must be equal to 1

If the two rv X and Y are independent, the conditional distribution is identical to the corresponding marginal distribut

The mle theta is either exactly or at least approximately the MVUE of theta

Two rv are said to be dependent if for every pair of x and y values, f(x,y) = f_xx * f_yy

The variance of a random variable X is defined to be Cov( X,X )

p is a measure of the degree of linear relationship between the variables X and Y

If X is a continuous random variable with pdf f(x) and cdf F(x), then at every x ... F'(x) = f(x)

The probability distribution of a statistic is referred to as its sampling distribution to emphasize that it describes h

The sampling of Xbar is centered precisely at the mean of the population from which the sample has been selected

The empirical rule states that if the pop distribution of a variable is approx sym, the roughly 95% are w/in 2 SD of mea

The parameter Beta in the gamma distribution is a scale parameter and alpha is shape parameter

The exponential distribution is frequently used as a model for the distribution of times between the occurence of succes

The standard gamma distribution has alpha = 1

z_alpha will denote the value on the z axis for which alpha of the area under the z-curves lies to the left of z_alpha

The symbol theta is customarily used to denote both the estimator of the parameter and the point estimate resulting from

Let X be a continuous rv with pdf f(x) and cdf F(x). Then for any number a, P(X<a) = 1-F(a)

The sampling distribution of Xbar has spread precisely the same as the spread of the population from which the sample ha

For any two random variables X and Y, -1 < p < 1

When X is a binomial rv with parameters n and p, the sample proportion p = X/n is an unbiased estimator of p

A good point estimator will have a small MSE

A point estimator theta is said to be a biased estimator of theta if E( theta ) = theta for every possible value of thet

If the distribution on which a probability plot is based is correct, the points in the plot should fall close to quad cu

We can use the normal distribution to approximate the binomial distribution by using a continuity correction of 1

For a uniform random variable, the density function is equal to 0 or 1

For continuous rv X and Y, P[(X,Y) c A)] is the volume underneath this surface and above the region A

The standard normal distribution is the normal distribution with parameters μ=12 and σ=0

The defect of covariance is that its computed value depends critically on the units of measurement

The mean square error is denoted by the MSE

Correlation is unaffected by shifts to either variable

It is frequently the case that just one probability plot will suffice for assessing the plausibility of an entire family

For continuous random variables, we replace summations in the definitions for discrete random variables with integration

The probability that the continuous random variable X takes on a value in the interval [a,b] is the area above this inte

The joint probability mass function p(x,y) for discrete rv X and Y is defined for each pair of numbers (x,y) by p(x,y)

For any alpa > 1, Gamma (alpha) = (alpha - 1)!

When two rv are not independent, there is no need to assess how strongly they are related to one another

The exponential distribution is a special case of both the gamma distribution and the Weibull distribution

The Central Limit Theorem can generally be used if n > 20

Gamma(1/2) = sqrt(pi)

The MVUE is the most likely among all unbiased estimators to produce an estimate close to the true theta

A random variable is continuous if only if the possible values form a single interval on the number line

The likelihood function tells us how likely the observed sample is as a function of the possible parameter values

Two random variable X and Y are said to be uncorrelated when p = 1

An effective way to check a distributional assumption is to correct what is called a probability plot

A value of p near 1 implies that increasing the value of X causes Y to increase