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Understanding Probability Distributions: Discrete vs. Continuous Variables

This comprehensive guide delves into the differences between discrete and continuous random variables in probability theory. Learn about their characteristics, probability distributions, and how to interpret them. Explore concepts like mean, sample space, uniform and normal density curves, and laws of large and small numbers. Discover how to analyze probability histograms and normal distributions effectively. Gain insights into calculating expected values, variances, and standard deviations of random variables.

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Understanding Probability Distributions: Discrete vs. Continuous Variables

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  1. Chapter 7 By: Sam Stone Kristy Fan Sophia Premji

  2. Random Variable • Variable whose value is a numerical outcome of a random phenomenon. • Usually denoted by a capital letter near the end of the alphabet. • Mean ( ): the average of the data. • Sample Space (S): list the possible values of the random variable. • S= [all numbers X such that 0 ≤ X ≤1] • P(S) = 1

  3. Overview: Discrete v. Continuous Discrete Continuous Takes all values in an interval of numbers Probability distribution is described by a density curve. X= amount of ____________ • Countable number of possible values • Probability distribution is described by a table of its values and probabilities. • X= number of _________

  4. Continuous Random Variable • Infinitely many possible values • Assigning probabilities for intervals of the values • Probabilities of continuous random variable are described by areas under density curves and values of x that make up the event. • Uniform density curves • Normal density curves • Assign probability 0 to every individual outcome because A=bh, and b=0 • Only intervals have positive probability • No distinction between ≥ or >

  5. Probability Histograms • Show probability distributions as well as distributions of data • Compare probability model for random digits: the height of each bar shows probability of outcome at base • All bars have the same width (areas of bars are proportional to their probabilities) • The height of all the bars should add up to 1 • Easy to compare two different distributions

  6. Normal Distribution • A type of probability distribution • N( • = mean • = standard deviation • N= normally distributed

  7. Law of Large numbers • Sampling distributions = probability distributions of random variables. • = mean of sample • is not exactly equal to 𝝁 • Different SRS yields different • If observations are continuously added to the random sample, will approach 𝝁. • Describes regular behavior of chance phenomenon in the long run.

  8. Law of small numbers • Does not actually exist • No regular pattern in a small number of trials. • Intuition does a poor job of distinguishing random behavior from systematic influences. • To simulate many trials on the calculator:2nd > list > seq(equation of pattern, variable used, start, end)

  9. Rules for meows • Mean of probability distribution (𝝁x) = expected value of X. • Mean of discrete random variable (where a, b=constant; X,Y=random variable) • 𝝁x = • 𝝁a+bx = a+ b𝝁x • 𝝁x+Y= 𝝁x + 𝝁Y

  10. Rules for Variances • Variance of discrete random variable (where a,b=constant; X,Y=independent random variables) • Standard deviation • Standard deviations do not add

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