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Chapter 7

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

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Chapter 7

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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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