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Probability Quiz Open Ended IE 415
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What is the notational convention for random variables and realizations of random variables?
Random variables are upper case, and realizations are lower case. X=the distance to the next gas station, x=50km
Name the three different probability functions and the type of random variables to which they apply
PDF: Probability density function, continuous. PMF: Probability mass function, discrete. CDF: Cumulative distribution function, discrete and continuous.
What are the functional notations used for the functions in the prior question?
PDF: f(x)= probability density at X=x, PMF: p(x)= probability that X=x, CDF: F(x)=probability that X<or=x
What are the mathematical properties that the probability functions must satisfy?
PDF: f(x) >or= 0, integral to -/+ infinity of f(x)dx = 1; PMF: p(x)<or= 1, p(x) >or= 0, sum of p(x) = 1; CDF: F(x)<or= 1, F(x) >or= 0, non-decreasing, lim over x to infinity F(x)=1
How are the two probability functions used to characterize discrete random variables mathematically related? Continuous?
Discrete: F(x) = sum of y <or= x P(y)
Continuous: F(x) = integral of neg infinity to x f(y)dx
For a random variable X what is the definition if the expected value E[X] and variance V[X] when X is discrete and continuous?
discrete E[X] = sum of p(x)
x, continuous E[X] = integral of -/+ infinity x
f(x)dx
discrete V[X] = sum[(x-E[X])^2]
p(x), continuous V[X] = integral -/+ infinity [(x-E[X])^2]
f(x)dx
What is a probability model? Give an example
A probability distribution used to represent some random quantity.
i.e. Time until next car accident distributed as exponential (2 hrs)
Why do some probability distributions have names?
These are probability distribution functions commonly used as probability models.
What is a statistic?
A value computed from data/numbers
What is a sampling distribution?
It is a probability distribution of a statistic.
For a normally distributed random variable X, with a mean u and a standard deviation q, what are the mean and standard deviation of the sample average of n observations of X? What is the distribution of the sample average of n observations?
Let ú=sample average of n observations of X. E[X] = u, stddev[ú] = q/√n
X is normal
What are two graphical methods for validating that observed data are from a normal distribution?
Normal probability plot, histogram
Given a set of numerical data, name two statistics that are measures of central tendency, and two statistics that are measures of unpredicatability
Central tendency: sample median, sample average
Unpredictability: sample stddev, range
What is the coefficient of variation for a random variable X? For n observations of X, how would you estimate the coefficient of variation of X?
CV(X) = stddev/mean
estimate CV(X) = sample stddev/sample mean
In statistical inference what does the null hypothesis represent?
Burden of proof is to reject null hypothesis, which represents the current or assumed situation.
What are the two types of errors in statistical interference? Explain what they mean.
Type I: Rejecting the null hypothesis when it is true (convicting the innocent)
Type II: Failing to reject the null hypothesis when it is false (freeing the guilty)
In a hypothesis test what are the three parameters that affect Type II error, and how does Type II error change as each parameter gets larger.
1. alpha = type I error level. As alpha decreases, beta (type II) increases
2. delta = difference to detect from the null hypothesis. As delta increases, beta decreases.
3. n = number of observations. As n increases, beta decreases.
What is the name of the set of graphs that show the relationship between type II error, sample size, and normalized difference to detect for a specific hypothesis test?
Operating characteristic curves (O.C.C.)
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