STAT 311 Final

• What are Degrees of Freedom?

  The number of independent values that can vary without breaking/contradicting analysis
• What does a Chi-squared random variable estimate?

  Variance
• What is a sample mean?

  Estimator for the mean $\mu$of a population
• The chi-squared distribution is the distribution of the sum of k i.i.d. squared standard normal variables.
• For any distribution with mean and variance, averaging independent draws from this distribution will produce a normal curve.
• Markov’s Inequality $P(X \geq a) \leq \frac{E[X]}{a}$

  gives upper bound on the upper tail probability of a random variable.
• Central Limit Theorem (CLT) says that for a big enough sample n, $\bar{X}$ can be approximated by a normal distribution.
• The closer the distribution of the Xi is to normal, the smaller n is required for the CLT.
• In most real-world data, n = 30 is a safe cutoff for CLT
• Use CLT to approximate P(Xbar <= x) for any x.Steps

  1. Find the mean and var for the distribution of Xbar
  2. Approximate the probability as a normal
  3. Find the Z score P(Z <= x - mu/ sd)
• A discrete random variable is a type of random variable that has a countable number of distinct values, such as heads or tails, playing cards, or the sides of a die. A continuous random variable can reflect an infinite number of potential values, such as the average rainfall in a region.
• The multinomial process generalizes

  the binomial
• Multinomial Process: The multinomial process is used when there are more than two possible outcomes, with probabilities p_1, ... , p_k. It involves selecting k items from a set of n items where each item can have one of k different types. This process allows us to calculate the probability of getting exactly m_1 items of type 1, m_2 items of type 2, etc., given the total number of selections made.
• Conditional Pmf of Y given X

  $P(Y=y|X=x) =$$\frac{P(X=x, Y=y)}{P(X=x)}$

  Takes two inputs X and Y

  Outputs the probability Y takes on the value y, given X can realized to a specific x

  The numerator is the joint pmf of X and Y

  The denominator is the marginal pmf of X

  Joint divided by the marginal of whatever variable we are conditioning on
• $P(X=x, Y=y) =$$P(Y=y|X=x) \cdot P(X=x)$

  The joint pmf is the product of the marginal and conditional pmfs.
• Continuous conditional pdf of Y given X

  $f_{Y|X}(y|x) =$$\frac{f_{XY}(x,y)}{f_X(x)}$

  The equation is the joint density divided by the marginal density
• The t-distribution is a family of distributions that arise when estimating the mean of a normally distributed population with unknown variance, particularly when the sample size is small
• There are two forms of the law of large numbers: the weak law which states that the sample mean converges in probability to the population mean, and the strong law which states that the sample mean converges almost surely to the population mean.
• A fair six-sided die is rolled repeatedly. What does the law of large numbers predict about the average value of the outcomes as the number of rolls increases?

  The law of large numbers predicts that as the number of rolls of a fair six-sided die increases, the average value of the outcomes will converge to the expected value of the die, which is$\frac{1+2+3+4+5+6}{6} =$$3.5$
• The Poisson distribution is a discrete probability distribution that describes the number of successes or failures in a sample.
• How many different ways... can be answered with a combination
• What is the probability of getting exactly k successes out of n trials if the probability of success on any one trial is p? This is called the binomial distribution