real number

Examples of real number in the following topics:

• Summary of the Uniform and Exponential Probability

  • X = a real number between a and b (in some instances, X can take on the values a and b). a = smallest X ; b = largest X
  • X = a real number, 0 or larger. m = the parameter that controls the rate of decay or decline

• Types of Variables

  • A variable is any characteristic, number, or quantity that can be measured or counted.
  • A variable is any characteristic, number, or quantity that can be measured or counted.
  • Numeric variables have values that describe a measurable quantity as a number, like "how many" or "how much. " Therefore, numeric variables are quantitative variables.
  • Observations can take any value between a certain set of real numbers.
  • Examples of discrete variables include the number of registered cars, number of business locations, and number of children in a family, all of of which measured as whole units (i.e., 1, 2, 3 cars).

• Practice 1: Goodness-of-Fit Test

  • The following data are real.
  • The cumulative number of AIDS cases reported for Santa Clara County is broken down by ethnicity as follows: (Source: HIV/AIDS Epidemiology Santa Clara County, Santa Clara County Public Health Department, May 2011)
  • If the ethnicity of AIDS victims followed the ethnicity of the total county population, fill in the expected number of cases per ethnic group.

• Are Real Dice Fair?

  • A fair die has an equal probability of landing face-up on each number.
  • A die (plural dice) is a small throw-able object with multiple resting positions, used for generating random numbers.
  • An example of a traditional die is a rounded cube, with each of its six faces showing a different number of dots (pips) from one to six.
  • Thus, they are a type of hardware random number generator.
  • All such dice are stamped with a serial number to prevent potential cheaters from substituting a die.

• Sampling Distributions and the Central Limit Theorem

  • Imagine rolling a large number of identical, unbiased dice.
  • Since real-world quantities are often the balanced sum of many unobserved random events, the central limit theorem also provides a partial explanation for the prevalence of the normal probability distribution.
  • The central limit theorem has a number of variants.
  • $n$ is the number of values that are averaged together not the number of times the experiment is done.
  • The sample means are generated using a random number generator, which draws numbers between 1 and 100 from a uniform probability distribution.

• Mean, Variance, and Standard Deviation of the Binomial Distribution

  • The easiest way to understand the mean, variance, and standard deviation of the binomial distribution is to use a real life example.
  • Consider a coin-tossing experiment in which you tossed a coin 12 times and recorded the number of heads.
  • If you performed this experiment over and over again, what would the mean number of heads be?
  • Therefore, the mean number of heads would be 6.
  • In general, the mean of a binomial distribution with parameters $N$ (the number of trials) and $p$ (the probability of success for each trial) is:

• What Does the Law of Averages Say?

  • While there is a real theorem that a random variable will reflect its underlying probability over a very large sample (the law of large numbers), the law of averages typically assumes that unnatural short-term "balance" must occur.
  • In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times.
  • It is important to remember that the law of large numbers only applies (as the name indicates) when a large number of observations are considered.
  • While different runs would show a different shape over a small number of throws (at the left), over a large number of rolls (to the right) they would be extremely similar.
  • Evaluate the law of averages and distinguish it from the law of large numbers.

• Conclusion

  • Many distributions in real life can be approximated using normal distribution.
  • In a probability histogram, the height of each bar shows the true probability of each outcome if there were to be a very large number of trials (not the actual relative frequencies determined by actually conducting an experiment).
  • We study the normal distribution extensively because many things in real life closely approximate the normal distribution, including:

• Practice 2: Confidence Intervals for Means, Unknown Population Standard Deviation

  • The following real data are the result of a random survey of 39 national flags (with replacement between picks) from various countries.
  • We are interested in finding a confidence interval for the true mean number of colors on a national flag.
  • Let X = the number of colors on a national flag.
  • Construct a 95% Confidence Interval for the true mean number of colors on national flags.

• The Gauss Model

  • The normal (Gaussian) distribution is a commonly used distribution that can be used to display the data in many real life scenarios.
  • Normal distributions are extremely important in statistics, and are often used in the natural and social sciences for real-valued random variables whose distributions are not known.
  • One reason for their popularity is the central limit theorem, which states that, under mild conditions, the mean of a large number of random variables independently drawn from the same distribution is distributed approximately normally, irrespective of the form of the original distribution.
  • Another reason is that a large number of results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically, in explicit form, when the relevant variables are normally distributed.
  • The normal distribution is symmetric about its mean, and is non-zero over the entire real line.