MATH30-6-Lecture-3 - Discrete Random Variables and...

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MATH30-6 Probability and Statistics Discrete Random Variables and Probability Distributions
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Objectives At the end of the lesson, the students are expected to Determine probabilities from the probability mass functions and the reverse; Determine probabilities from cumulative distribution functions and cumulative distribution functions from probability mass functions, and the reverse; Calculate the means and variances for discrete random variables; Understand the assumptions for some common discrete probability distributions;
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Objectives At the end of the lesson, the students are expected to Select an appropriate discrete probability distribution to calculate probabilities in specific applications; and Calculate probabilities, determine means and variances for some common discrete probability distributions.
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Probability Mass Function For a discrete random variable X with possible values x 1 , x 2 , …, x n , a probability mass function is a function such that (1) f ( x i ) ≥ 0 (2) (3) f ( x i ) = P ( X = x i ) (3-1)
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Probability Mass Function Examples: 3-4/68 Digital Channel There is a chance that a bit transmitted through a digital transmission channel is received in error. Let X equal the number of bits in error in the next four bits transmitted. The possible values for X are {0, 1, 2, 3, 4}. Based on a model for errors that is presented in the following section, probabilities for these values will be determined. Suppose that the probabilities are P ( X = 0) = 0.6561 P ( X = 1) = 0.2916 P ( X = 2) = 0.0486 P ( X = 3) = 0.0036 P ( X = 4) = 0.0001
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Probability Mass Function The probability distribution of X is specified by the possible values along with the probability of each. A graphical description of the probability distribution of X is shown on Fig. 3-1. Practical Interpretation: A random experiment can often be summarized with a random variable and its distribution. The details of the sample space can often be omitted.
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Probability Mass Function
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Probability Mass Function 3-5/68 Wafer Contamination 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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Probability Mass Function 3-16/69 The sample space of a random experiment is { a , b , c , d , e , f }, and each outcome is equally likely. A random variable is defined as follows: Determine the probability mass function of X . Use the probability mass function to determine the following probabilities: (a) P ( X = 1.5) (b) P (0.5 < X < 2.7) (c) P ( X > 3) (d) P (0 ≤ X < 2) (e) P ( X = 0 or X = 2) outcome a b c d e f x 0 0 1.5 1.5 2 3
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Probability Mass Function For Exercises 3-17 to 3-21, verify that the following functions are probability mass functions, and determine the requested probabilities.
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