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ENGG2012B Lecture 18 Random variable

ENGG2012B Lecture 18 Random variable. Kenneth Shum. Histogram of midterm score. The idea of random variables.

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ENGG2012B Lecture 18 Random variable

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  1. ENGG2012BLecture 18Random variable Kenneth Shum ENGG2012B

  2. Histogram of midterm score ENGG2012B

  3. The idea of random variables • Given a sample space  and a probability measure defined on it, very often we cannot observe the elements in the sample space directly, but through a function defined on the sample space. • For each , we observe a real number X(). This is a function of . • We would like to calculate the probability of the events such as X()  0, X() = 30, etc. ENGG2012B

  4. Formal definition of RV • A random variable is not a variable. It is a function from the sample space to the real numbers. • There is nothing random in the function itself. The randomness comes from the probability measure assigned to the events of the sample space. • A random variable is often written as X(), where  is an element in the sample space.   The real line X() ENGG2012B

  5. Example: Midterm score The “midterm score” is a random variable. It maps a student to the corresponding midterm score. Student C 35 Student A 23 Student B Class of ENGG2012B Real line If we randomly pick a student out ofthe total of 90 students, the probabilitythat “midterm score = 23” is 2/90. ENGG2012B

  6. Example: dart Note: the associationof a point on the dartboardand the score is completelydeterministic. The randomnessonly comes from throwingthe random dart. • In the random experiment of rowing a dart randomly to a circle of radius 3, let X() be the score defined by • X() = 3 if the distance to the center is less than 1. • X() = 2 if the distance to the center is between 1 and 2. • X() = 1 if the distance to the center is between 2 and 3. dartboard 3 2 Pr( X() = 3) = 12 / (32) = 1/9 Pr(X() = 2) = ( 22 –12)/ (32) =1/3 1 Pr(X() = 1) = ( 32 –22)/ (32)=5/9 ENGG2012B

  7. Notations • The probability function Pr is sometime regarded as a operator, which takes an event as input and outputs a real number. • If we want to emphasize that “Pr” is a function with events as input, we shall write for the probability of X() = 1, or simply ENGG2012B

  8. Probability distribution function • Sometimes, we underlying sample space  is very complicated or very large. We only care about the distribution of the random variable, but not sample space . • We may try to forget the sample space and work with the probability mass function (pmf), f(i) = Pr(X() = i). • If we want to emphasize that it is the pmf of random variable X(), we can write fX(i). • The pmf is sometimes called the “distribution of X”. However, in probability, the term “distribution” usually refers to something else, namely, F(z) = Pr(X()  z). • In the remainder of this lecture, we assume that the random variables take values in the integers, i.e., i is an integer. ENGG2012B

  9. Example: sum of two dice • Roll two fair dice. Let X() be the sum. The pmf of X() can be tabulated as follows. The sample spaceconsists of 36 outcomes ENGG2012B

  10. Example: number of heads • Toss 5 fair coins. Let Y() be the number of heads. The pmf of Y() in tabular form is ENGG2012B

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