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Random Variables

Random Variables. Example: We roll a fair die 6 times. Suppose we are interested in the number of 5’s in the 6 rolls. Let X = number of 5’s. Then X could be 0, 1, 2, 3, 4, 5, 6. X = 0 corresponds to the 56 elements of our 66 elements of Ω. X = 1 corresponds to the elements etc.

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Random Variables

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  1. Random Variables • Example: We roll a fair die 6 times. Suppose we are interested in the number of 5’s in the 6 rolls. Let X = number of 5’s. Then X could be 0, 1, 2, 3, 4, 5, 6. X = 0 corresponds to the 56 elements of our 66 elements of Ω. X = 1 corresponds to the elements etc. X is an example of a random variable. • Probability models often stated terms of random variables. E.g. - model for the # of H’s in 10 flips of a coin. - model for the height of a randomly chosen person. - model for size of a queue. week 3

  2. Discrete Probability Spaces (Ω, F, P) week 3

  3. Discrete Random Variable • Definition: A random variable X is said to be discrete if it can take only a finite or countably infinite number of distinct values. • A discrete random variable X maps the sample space Ω onto a countable set. Define a probability mass function (pmf) or frequency function on X such that Where the sum is taken over all possible values of X. • Note that there is a theorem that states that there exists a probability triple and random variable whenever we have a function p such that • Definition: The probability distribution of a discrete random variable X is represented by a formula, a table or a graph which provides the list of all possible values that X can take and the pmf for each value week 3

  4. Examples of Discrete Random Variables • Discrete Uniform Distribution We roll a fair die. Let X = the # that comes up. We have that This is an example of equiprobable outcomes, that is To state the probability distribution of X we need to give its possible values and its pmf X is a discrete Uniform random variable. X has a uniform distribution. week 3

  5. Bernoulli Distribution week 3

  6. Binomial Distribution • Roll a die n time and count the number of times 6 came up. Let X be the number of 6’s in n rolls. X has image {1, 2, …, n} The probability distribution of X is given by the following formula • In general, if identical Bernoulli trail is repeated n times independently and X is a random variable that count the number of success in the n trails then the probability distribution of X is given by Where p is the probability of success on any one experiment. X is a Binomial random variable. X has a Binomial Distribution. • Question: is this a valid pmf? Prove! week 3

  7. Geometric Distribution • We roll a fair die until the first 6 comes up. Let X = the number of rolls until we get the first 6. Possible values of X: {1, 2, 3, …..} The probability distribution of X is given by the following formula • In general, if identical Bernoulli trail is repeated independently until the first success is obtained and X is a random variable that count the number of trials until the first success then the probability distribution of X is given by X is a Geometric random variable. X has a Geometric Distribution. • Question: is this a valid pmf? Prove! week 3

  8. In general for a Geometric distribution: • Memory-less property of geometric random variable: for i > j week 3

  9. Negative Binomial Distribution • We roll a fair die until the second 6 comes up. This is the waiting time for the second 6. Let X = the number of rolls until we get two 6’s. Possible values of X: {2, 3, 4, …..} The probability distribution of X is given by the following formula • Is this a valid pmf? Prove! • In general, X is the total number of experiments when waiting for rth success in a sequence of independent Bernoulli trails. The probability distribution of X is given by X has a Negative Binomial random Distribution. week 3

  10. Hypergeometric Distribution • A hat contains 12 tickets, 7 black and 5 white. Three tickets are drawn at random. Let X = the # of black tickets drawn. X could be 0, 1, 2, 3. The probability mass for each value can be calculated using combinatorics. For example, week 3

  11. Poisson Distribution • Model for the number of events occurring in a time (or space) interval where λ (a parameter of the distribution) is the rate of the occurrence of the events per one unit of time (or space). • A Poisson random variable X = number of events per one unit of time (space). Possible values for X: {0, 1, 2, … } The probability distribution of X is given by • Is this a valid pmf? Prove! week 3

  12. Distribution Function of Random Variables • Definition A cumulative distribution function (cdf) of a random variable X is a mapping F: R [0, 1] defined by • If X is a discrete random variable with pmf for x = 0, 1, 2, … then where is the greatest integer ≤v. • Example: week 3

  13. Properties of Distribution Function • F is monotone, non decreasing i.e. F(x) ≤F(y) if x≤y. • As x- ∞ , F(x)  0 • As x∞, F(x)  1 • F(x) is continuous from the right • For a < b Why? week 3

  14. Exercises • A box contain 20 notes numbered 20 to 39. We randomly pick one note and record its number. What is the probability that the number we got is greater then 32? • 30% of U of T students wear glasses. We select a random sample of size 10 students. a) What is the probability that exactly 4 of them wear glasses? b) What is the probability that more then 3 wear glasses? • We roll a die until we obtained an even outcome. a) What is the probability that we will roll the die exactly 5 times? b) What is the probability that we roll the die more then 7 times ? c) What is the probability that we roll the die more then 7 times if we know that we need more then 2 rolls? week 3

  15. We roll a die until we get 6 even outcomes. a) What is the probability that we need exactly 10 rolls? b) What is the probability that we need less 10 rolls? • The number of cars that cross Spadina and Bloor intersection is a Poisson random variable with λ = 15 cars per minute. a) What is the probability that in a given minute exactly15 cars will cross the intersection? b) What is the probability that in a given minute more then 15 cars will cross the intersection? c) What is the probability that during half an hour there where exactly 2 minutes in which 15 cars crossed the intersection? week 3

  16. Relation between Binomial and Poisson Distributions • Binomial distribution Model for number of success in n trails where P(success in any one trail) = p. • Poisson distribution is used to model rare occurrences that occur on average at rate λ per time interval. Can think of “rare” occurrence in terms of p 0 and n ∞. Take these limits so that λ = np. • So we have that week 3

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