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Random Variable

Random Variable. A numerical measure of the outcome of the probability experiment whose value is determined by chance. Random variables. Typically denoted by capital letters, such as X. X= 0, 1, or 2 for the number of heads in two flips of a coin.

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Random Variable

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  1. Random Variable A numerical measure of the outcome of the probability experiment whose value is determined by chance.

  2. Random variables • Typically denoted by capital letters, such as X. • X= 0, 1, or 2 for the number of heads in two flips of a coin. • Lowercase letter is used to list the sample space of the experiment.

  3. 2 types of Random Variables • Discrete - finite or countable number of values • Continuous- has infinitely many values.

  4. Probability Distribution Table, Graph, or Math Formula • Provides the possible values of the random variable and their corresponding probabilities.

  5. The area of each rectangle … • In a prob. Histogram, the area of each rectangle is equal to the prob that the random variable assumes the particular value.

  6. The mean of a discrete random variable is … • The sum of all values… • X times the P(x) where x is the value of the random variable and P(x) is the probability of observing the value of x.

  7. Round the mean variance and standard deviation to … • One more decimal place than the values of the random variable.

  8. Interpretation of the mean of a discrete random variable… • Suppose that an experiment is repeated n independent times and the value of the random variable X is recorded. • As the number of repetitions of the experiment increases, the mean value of the n trials will approach the mean of the random variable X.

  9. In other words… • Let x sub 1 be the value of the random variable X after the 1st experiment, and so on. Then • X bar= (x sub 1 +x sub2 +….+ x sub n) / n

  10. Mean is also called • The expected value E(x). • Expected value plays a role in game theory. For example a negative sum game has an expected value that is negative.

  11. Variance and Standard Deviation • The sum of (x – mean ) squared times the prob of X. • X=the value of the random variable , mu sub x is the mean and P(x) is the prob of observing the variable X.

  12. Computing variance and stand dev: • Remember the variance is s squared in a sample and the standard deviation is the square root of s.

  13. Some example problems….

  14. Last section 6-2 • Criteria for binomial experiment • 1. the experiment is performed a fixed number of times…called trials. • 2. the trials are independent. • 3. for each trial, there are only 2 mutually exclusive outcomes…successes or failures. • 4. the prob of success is the same for each trial of the experiment.

  15. Notation used… • n is the number of trials. • P is the probability of success. • 1-p is the prob of failure…like the complement. • X is the number of successes in n independent trials 0 is less than or equal to x which is less than or equal to n.

  16. Binomial Probability Distribution Function • P(x)= subn C sub x times p to the x power times (1-p) to the (n-x) power.

  17. To do a prob dist on the calculator, • Go to 2ndVars then gotobinompdf, then type in n, p, x. This will give you the Probability of a binomial distribution.

  18. Mean and Stand Dev of Binomial Random Variable: • Mu sub x = n times p where n is the number of trials and p is the prob. Of success. • Standard dev = the square root of (n times p times(1-p).

  19. Review for test unit 2 • Pages 324-325 1-21 odd and pages 368 1-4, 6, 7

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