probability

Examples of probability in the following topics:

• Marginal and joint probabilities

  • For instance, a probability based solely on the student variable is a marginal probability:
  • A probability of outcomes for two or more variables or processes is called a joint probability:
  • If a probability is based on a single variable, it is a marginal probability.
  • Verify Table 2.14 represents a probability distribution: events are disjoint, all probabilities are non-negative, and the probabilities sum to 1.24.
  • We can compute marginal probabilities using joint probabilities in simple cases.

• Misconceptions

  • State why the probability value is not the probability the null hypothesis is false
  • Misconception: The probability value is the probability that the null hypothesis is false.
  • Proper interpretation: The probability value is the probability of a result as extreme or more extreme given that the null hypothesis is true.
  • It is the probability of the data given the null hypothesis.
  • It is not the probability that the null hypothesis is false.

• Probability

  • Probability density function describes the relative likelihood, or probability, that a given variable will take on a value.
  • In probability theory, a probability density function (pdf), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value.
  • The probability for the random variable to fall within a particular region is given by the integral of this variable's probability density over the region.
  • For a continuous random variable $X$, the probability of $X$ to be in a range $[a,b]$ is given as:
  • Apply the ideas of integration to probability functions used in statistics

• Chi-Square Probability Table

• Probability Distributions for Discrete Random Variables

  • The probability distribution of a discrete random variable $x$ lists the values and their probabilities, where value $x_1$ has probability $p_1$, value $x_2$ has probability $x_2$, and so on.
  • Every probability $p_i$ is a number between 0 and 1, and the sum of all the probabilities is equal to 1.
  • $\sum f(x) = 1$, i.e., adding the probabilities of all disjoint cases, we obtain the probability of the sample space, 1.
  • Sometimes, the discrete probability distribution is referred to as the probability mass function (pmf).
  • The probability mass function has the same purpose as the probability histogram, and displays specific probabilities for each discrete random variable.

• Continuous Probability Distributions

  • A continuous probability distribution is a probability distribution that has a probability density function.
  • Each of these individual outcomes has probability zero, yet the probability that the outcome will fall into the interval (3 cm, 4 cm) is nonzero.
  • Unlike a probability, a probability density function can take on values greater than one.
  • The standard normal distribution has probability density function:
  • Boxplot and probability density function of a normal distribution $$$N(0, 2)$.

• Student Learning Outcomes

• Probability distributions

  • A probability distribution is a table of all disjoint outcomes and their associated probabilities.
  • A probability distribution is a list of the possible outcomes with corresponding probabilities that satisfies three rules:
  • Probability distributions can also be summarized in a bar plot.
  • 2.20: The probabilities of (a) do not sum to 1.
  • The second probability in (b) is negative.

• Two Types of Random Variables

  • Their probability distribution is given by a probability mass function which directly maps each value of the random variable to a probability.
  • For example, the value of $x_1$ takes on the probability $p_1$, the value of $x_2$ takes on the probability $p_2$, and so on.
  • The probabilities $p_i$ must satisfy two requirements: every probability $p_i$ is a number between 0 and 1, and the sum of all the probabilities is 1.
  • The resulting probability distribution of the random variable can be described by a probability density, where the probability is found by taking the area under the curve.
  • This shows the probability mass function of a discrete probability distribution.

• Common Discrete Probability Distribution Functions

  • Some of the more common discrete probability functions are binomial, geometric, hypergeometric, and Poisson.
  • A probability distribution function is a pattern.
  • You try to fit a probability problem into a pattern or distribution in order to perform the necessary calculations.
  • These distributions are tools to make solving probability problems easier.