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2. Discrete Random Variables and Distributions

2. Discrete Random Variables and Distributions. A random variable assigns a real number to every possible outcome or event in an experiment. Random variables are normally represented by a capital letter such as X, Y, or Z.

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2. Discrete Random Variables and Distributions

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  1. 2. Discrete Random Variables and Distributions

  2. A random variable assigns a real number to every possible outcome or event in an experiment. Random variables are normally represented by a capital letter such as X, Y, or Z. A random variable is either (a) discrete if it can assume only a finite or limited number of possible values; or (b) continuous if it has an unlimited or finite set of possible values.

  3. Discrete Probability Distributions A discrete probability distribution lists or shows all possible values of a random variable, along with the probability of each possible value. Example: roll one die Written as a table: Displayed as a graph:

  4. Discrete Probability Distributions A discrete probability distribution lists or shows all possible values of a random variable, along with the probability of each possible value. Notice the probabilities always sum to 1.00 Example: Age of students in class. Displayed as a graph: Written as a table:

  5. Distribution Statistics The Expected Value or Mean of a distribution is the (weighted) average of all possible outcomes. If X denotes the random variable, the expected value is written either as X or E[X]. It is calculated as… E[X] = Ʃpr(Xi)Xi (multiply each possible outcome by its probability, and add them all up)

  6. Other Statistics Identifying the “Center” of the Distribution The Mode is the outcome that has the most observations or highest probability of occurring. The Median is the outcome that has an equal number of outcomes greater than, and less than, that outcome. Note: When a probability distribution is symmetrical, the mean, median and mode will all have the same value.

  7. Student Age Example (revisited) The average age E[X] = (0.10 x 18) + (0.15 x 19) + (0.30 x 20) + (0.40 x 21) + (0.05 x 23) = 20.20 years The mode is 21. The median age is 20 years.

  8. Statistics of Dispersion * How “spread out” is the distribution. The Variance is the (weighted) average of squared differences from the mean. The variance is usually denoted as Var(X) or The variance can be calculated as either The Standard Deviation is also used. It is just the square root of the variance.

  9. Student Age Example (revisited)

  10. Using MS Excel Statistical Functions….

  11. The Binomial Distribution(Bernoulli proces) • 2 Outcomes • Each trial has only two possible outcomes; e.g. heads/tails; boy/girl, etc. • Stable Probability • The probability p of “success” stays the same; as does the probability q = 1 – p or “failure”. • Independent Trials • The trials are statistically independent. • Finite Number • The number of trials n is a positive integer. r denotes the number of successful trials.

  12. Binomial Statistics Expected Value (mean) E[X] = np Variance var[X] = np(1 – p) =npq

  13. Binomial Distribution: Example 40% of students at a university are male. What is the probability of getting only 3 males in a class of 22 students? What are the distribution statistics? P = 0.40; q = 1 – 0.40 = .60; n = 22; r = 3 E[r] = np = 22x0.40 = 8.8 males; 13.2 females Var[r] = np(1 – p) = 22x0.40(1 – 0.40) = 5.28

  14. Binomial Distribution: Example cont. The probability of getting exactly 3 males in a class of 22 is 0.19 or 19%: The probability of getting exactly 3 males can also be found in a Binomial table (not shown).

  15. * Inappropriate for previous example.

  16. Continuous RVs and Distributions

  17. The Normal Distribution

  18. The Normal Distribution (Bell Curve)

  19. Standard Normal Curve It is often convenient to transform a normally distributed r.v. into a standard normal r.v. by using the following z-transformation: Where X is the underlying random variable with mean μ and standard deviation σ. Z will have a mean E[Z] = 0 and standard deviation σz = 1.

  20. How Standard Normal Distributions Are Used Example: random variable X has a normal distribution with μ = 100 and σ = 15. What is the probability, say, that X < 130? Answer: Calculate the Z-score of (130-100)/15 = 2 (i.e. 2 standard deviations). Then look on a standard normal table to get 0.97725.

  21. Example Illustrated

  22. Think About…. Standardizing normal variables into Z-scores allows us to calculate probabilities. We’ve seen one example for calculating a Z-score < some particular value. Question: How would you calculate the probability to get a score (a) above a particular value; or (b) between two particular values?

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