LECTURE CHAPTER 3 DESCRETE RANDOM VARIABLE

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1 LECTURE CHAPTER 3 DESCRETE RANDOM VARIABLE MSc Đào Việt Hùng hungdv@tlu.edu.vn

2 Random Variable A random variable is a function that assigns a real number to each outcome in the sample space of a random experiment. Notation A random variable is denoted by an uppercase letter such as X. After an experiment is conducted, the measured value of the random variable is denoted by a lowercase letter such as x = 70 milliamperes. Types of Random Variable A discrete random variable is a random variable with a finite (or countably infinite) range. A continuous random variable is a random variable with an interval (either finite or infinite) of real numbers for its range.

3 Examples of Random Variable The waiting time for a bus. The number of customers having dinner at the restaurant from 6 pm to 8 pm. The volume of gasoline that is lost to evaporation during the filling of a gas tank. The length of a randomly selected telephone call. The height of a student chosen randomly

4 Probability Distribution The probability distribution of a random variable X is a description of the probabilities associated with the possible values of X PROBABILITY MASS FUNCTION For a discrete random variable X with possible values x 1,..., x n, a probability mass function is a function such that f(x i ) 0 n f(x i ) = 0 i=1 f(x i ) = P (X = x i )

5 Example Random experiment: flipping a coin twice Let X = the number of heads Sample space S = {}{{} T T, T H, HT, }{{}}{{} HH} x=0 x=1 x=2 Possible values: x = 0, 1, 2 X is a discrete random variable with the probability distribution as follow f(0) = P (X = 0) = 1 4 ; f(1) = P (X = 1) = 2 4 f(2) = P (X = 2) = 1 4 Possible Values: x Probabilities: f(x) = P (X = x) 1/4 2/4 1/4

6 Example A shipment of 20 similar laptop computers to a retail outlet contains 3 that are defective. If a school makes a random purchase of 2 of these computers, find the probability distribution for the number of defectives?

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8 Example Let the random variable X denote the number of semiconductor wafers that need to be analyzed in order to detect a large particle of contamination. Assume that the probability that a wafer contains a large particle is 0.01 and that the wafers are independent. Determine the probability distribution of X?

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10 Example The probability distribution of X is given by x f(x) P (0 < X < 3)? P (X 2)? P (X > 2)?

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12 CUMULATIVE DISTRIBUTION FUNCTION The cumulative distribution function of a discrete random variable X, denoted as F (x), is F (x) = P (X x) = x i x f(x i ) For a discrete random variable X, F (x) satisfies the following properties. F (x) = P (X x) = f(x i ) x i x 0 F (x) 1 If x y then F (x) F (y)

13 Example Given the random variable X with the probability distribution is x f(x) Find the cumulative distribution of X Using F (x) to find P (X 1.2)?

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15 Example Suppose that a day s production of 850 manufactured parts contains 50 parts that do not conform to customer requirements. Two parts are selected at random, without replacement, from the batch. Let the random variable X equal the number of nonconforming parts in the sample. What is the CDF of X?

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17 MEAN AND VARIANCE The mean or expected value of the discrete random variable X, denoted as µ or E(X) is µ = E(X) = x xf(x) The variance of X, denoted as σ 2 or V (X), is σ 2 = V (X) = E(X µ) 2 = x x 2 f(x) µ 2 The standard deviation of X is σ = V (X)

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20 Example (Marketing) Two new product designs are to be compared on the basis of revenue potential. Marketing feels that the revenue from design A can be predicted quite accurately to be $3 million. The revenue potential of design B is more difficult to assess. Marketing concludes that there is a probability of 0.3 that the revenue from design B will be $7 million, but there is a 0.7 probability that the revenue will be only $2 million. Which design do you prefer?

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22 Example (Massages) The number of messages sent per hour over a computer network has the following distribution: x = # of massages f(x) Determine the mean and standard deviation of the number of messages sent per hour.

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24 Example (Binomial Distribition) Flipping a fair coin 4 times. Let X is the number of heads. a) Find the PMF for X? b) Determine the average number of heads and the variance?

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