random number

Examples of random number in the following topics:

• Lab 2: Sampling Experiment

  • In this lab, you will be asked to pick several random samples.
  • In each case, describe your procedure briefly, including how you might have used the random number generator, and then list the restaurants in the sample you obtained
  • Round to the nearest whole number.
  • Round to the nearest whole number.
  • The number of restaurants will vary.

• Two Types of Random Variables

  • A random variable $x$, and its distribution, can be discrete or continuous.
  • They may also conceptually represent either the results of an "objectively" random process (such as rolling a die), or the "subjective" randomness that results from incomplete knowledge of a quantity.
  • 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.
  • As a result, the random variable has an uncountable infinite number of possible values, all of which have probability 0, though ranges of such values can have nonzero probability.
  • Selecting random numbers between 0 and 1 are examples of continuous random variables because there are an infinite number of possibilities.

• Randomized Design: Single-Factor

  • In complete random design, the run sequence of the experimental units is determined randomly.
  • Because of the replication, the number of unique orderings is 90 (since $90=\frac{6!}
  • All completely randomized designs with one primary factor are defined by three numbers: $k$ (the number of factors, which is always 1 for these designs), $L$ (the number of levels), and $n$ (the number of replications).
  • The total sample size (number of runs) is $N=k\cdot L \cdot n$.
  • An example of a completely randomized design using the three numbers is:

• Probability Distributions for Discrete Random Variables

  • A discrete random variable $x$ has a countable number of possible values.
  • The number of eggs that a hen lays in a given day (it can't be 2.3)
  • The number of students that come to class on a given day
  • The number of people in line at McDonald's on a given day and time
  • For example, suppose that $x$ is a random variable that represents the number of people waiting at the line at a fast-food restaurant and it happens to only take the values 2, 3, or 5 with probabilities $\frac{2}{10}$,  $\frac{3}{10}$, and $\frac{5}{10}$ respectively.

• Expected Values of Discrete Random Variables

  • The expected value of a random variable is the weighted average of all possible values that this random variable can take on.
  • A discrete random variable $X$ has a countable number of possible values.
  • The expected value may be intuitively understood by the law of large numbers: the expected value, when it exists, is almost surely the limit of the sample mean as sample size grows to infinity.
  • In this case, since all outcomes are equally likely, we could have simply averaged the numbers together: $\frac{1+2+3+4+5+6}{6} = 3.5$.
  • An illustration of the convergence of sequence averages of rolls of a die to the expected value of 3.5 as the number of rolls (trials) grows.

• The Hypergeometric Random Variable

  • A hypergeometric random variable is a discrete random variable characterized by a fixed number of trials with differing probabilities of success.
  • As random selections are made from the population, each subsequent draw decreases the population causing the probability of success to change with each draw.
  • A random variable follows the hypergeometric distribution if its probability mass function is given by:

• Introduction

  • Continuous random variables have many applications.
  • The field of reliability depends on a variety of continuous random variables.
  • NOTE: The values of discrete and continuous random variables can be ambiguous.
  • For example, if X is equal to the number of miles (to the nearest mile) you drive to work, then X is a discrete random variable.
  • How the random variable is defined is very important.

• Probability

  • The probability of an outcome is the proportion of times the outcome would occur if we observed the random process an in finite number of times.
  • As the number of rolls increases, $\bar{\rho}_n$ will converge to the probability of rolling a 1, p = $\frac{1}{6}$ .
  • The tendency of $\bar{\rho}_n$ to stabilize around ${\rho}$ is described by the Law of Large Numbers.
  • However, these deviations become smaller as the number of rolls increases.
  • The outcome of this process will be a positive number.

• Summary

  • Quantitative Data (a number)- Discrete (You count it. )- Continuous (You measure it. )

• Random Variable Notation

  • Upper case letters like X or Y denote a random variable.
  • If X is a random variable, then X is written in words. and x is given as a number.
  • For example, let X = the number of heads you get when you toss three fair coins.
  • X is in words and x is a number.
  • Because you can count the possible values that X can take on and the outcomes are random (the x values 0, 1, 2, 3), X is a discrete random variable.