probability distribution

Examples of probability distribution in the following topics:

• Student Learning Outcomes

• Student Learning Outcomes

• Probability distributions

  • A probability distribution is a table of all disjoint outcomes and their associated probabilities.
  • Table 2.5 shows the probability distribution for the sum of two dice.
  • 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.
  • The probability distribution for the sum of two dice is shown in Table 2.5 and plotted in Figure 2.8.

• 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.
  • Each distribution has its own special characteristics.

• Continuous Probability Distributions

  • A continuous probability distribution is a representation of a variable that can take a continuous range of values.
  • A continuous probability distribution is a probability distribution that has a probability density function.
  • There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.
  • The standard normal distribution has probability density function:
  • Boxplot and probability density function of a normal distribution $$$N(0, 2)$.

• Probability Distributions for Discrete Random Variables

  • Probability distributions for discrete random variables can be displayed as a formula, in a table, or in a graph.
  • 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.
  • A discrete probability distribution can be described by a table, by a formula, or by a graph.
  • The formula, table, and probability histogram satisfy the following necessary conditions of discrete probability distributions:
  • Sometimes, the discrete probability distribution is referred to as the probability mass function (pmf).

• Properties of Continuous Probability Distributions

  • The graph of a continuous probability distribution is a curve.
  • The cumulative distribution function is used to evaluate probability as area.
  • There are many continuous probability distributions.
  • When using a continuous probability distribution to model probability, the distribution used is selected to best model and fit the particular situation.
  • In this chapter and the next chapter, we will study the uniform distribution, the exponential distribution, and the normal distribution.

• Marginal and joint probabilities

  • For instance, a probability based solely on the student variable is a marginal probability:
  • If a probability is based on a single variable, it is a marginal probability.
  • proportions in Table 2.13.The joint probability distribution of the parents and student
  • 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.

• Conclusion

  • Many distributions in real life can be approximated using normal distribution.
  • Simply looking at probability histograms makes it easy to see what kind of distribution the data follow .
  • A normal probability plot is a graphical technique for normality testing--assessing whether or not a data set is approximately normally distributed.
  • We study the normal distribution extensively because many things in real life closely approximate the normal distribution, including:
  • This probability histogram shows the probabilities that 0, 1, 2, 3, or 4 heads will show up on four tosses of a fair coin.

• History of the Normal Distribution

  • In the chapter on probability, we saw that the binomial distribution could be used to solve problems such as "If a fair coin is flipped 100 times, what is the probability of getting 60 or more heads?
  • " The probability of exactly x heads out of N flips is computed using the formula:
  • Therefore, to solve this problem, you compute the probability of 60 heads, then the probability of 61 heads, 62 heads, etc., and add up all these probabilities.
  • The heights of the blue bars represent the probabilities.
  • Note how well it approximates the binomial probabilities represented by the heights of the blue lines.