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Unit 6 – Chapter 6

Unit 6 – Chapter 6. 6.1 – Discrete and Continuous Random Variables. Do you remember?. What is a probability distribution?. Example: Flipping Out!. Below is a probability distribution for flipping a coin four times and counting how many heads we see:

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Unit 6 – Chapter 6

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  1. Unit 6 – Chapter 6 6.1 – Discrete and Continuous Random Variables

  2. Do you remember? • What is a probability distribution?

  3. Example: Flipping Out! • Below is a probability distribution for flipping a coin four times and counting how many heads we see: • The number of heads we see is our Random Variable. In this problem we called it “X.” • Remember that a distribution shows what is possible and how often.

  4. Definitions • A random variable takes numerical values that describes the outcomes of some chance process. • The probability distribution of a random variable gives its possible values and their probabilities.

  5. Discrete Random Variable • A discrete random variable X takes a fixed set of possible values with gaps between. • For example: grades on the AP test • (1, 2, 3, 4 ,5)

  6. Dice, Dice Everywhere! • Come grab a die from me. • Roll the die 20+ times, recording your results as you go • Find the average and standard deviation for your die rolls • Record your mean and standard deviation • Share your mean and standard deivation them on the board

  7. Expected Value (Mean) • The mean of a discrete random variable X is also an average of the possible values of X. • To find the mean (expected value) of X, multiply each possible value by its probability then add all of the products. • Note:

  8. Example: A Blast from the Past • Below is the score distribution for AP tests waaaaay back in 1997: • So we “expect” to see a score of 3.022. • Notice that expected value should NOT be rounded. Remember, this is an average!

  9. Press your luck: • On an American roulette wheel there are 38 slots numbered 1 through 38. Slots are numbered 1-36 (half red half black) and two extra slots are labeled 0 or 00 and are green. If a player places a $1 bet on “red” the probability distribution for their net gain is as follows: • Find the expected value for net gain.

  10. Standard Deviation (and Variance) of a Discrete Random Variable • When you use the mean (expected value) you will also use the standard deviation. • ) • Standard Deviation is the square root of the variance. • Note:

  11. Example: A Blast from the Past… Again • Below is the score distribution for AP tests waaaaay back in 1997… for a second time:

  12. Press your luck… Again: • Using the same roulette example, find the standard deviation for net gain, when betting $1 on red:

  13. Continuous Random Variable • A continuous random variable, X, takes all values in an interval of numbers. The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that make up the event. • For example – heights of all people in this room

  14. You decide! Discrete or Continuous? • Time it takes you to clean your room • How much it costs for dinner • Inches of rainfall last month • Your birthday • The outcome of the roll of a dice

  15. Homework • P. 353: 2-8 even, 9, 12-18 even, 21-30

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