4.1 Example_Class_3_Solution

4.1 Example_Class_3_Solution - THE UNIVERSITY OF HONG KONG...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I, FALL 2010 EXAMPLE CLASS 3 Random Variable Elements of Theory A function (with some requirement) random variable. - defined on the sample space { } is called a Domain is the sample space Range is usually a numbers set, e.g., or its subsets, for easy manipulation. The range is called the state space of the random variable. There is no intrinsic difference on the nature between a sample space and a state space—they are just two sets with some requirement, called “measurability.” They are just domain and range of a “function” with some requirement, called “measurability.” The variable perspective is adopted by an observer of a random experiment. The observer is only able to observe/know/measure/obtain information based on the state space. For the observer, all she could see is a variable dancing (randomly) on the state space. This is the perspective that we will primarily study in this course. The function perspective is adopted by someone who would have a “divine” capacity in understanding (a deterministic part of) the design of the random mechanism, in particular, her capacity in seeing the exist ence of an underlying sample space as the domain of a function sending elements of the domain to the state space. You will be studying this perspective in an advanced course of probability. There is a law governing how any random variable to be observed in its state space. The law is a probabilistic one, called the probability distribution of a random variable. There are two qualifications for any real-valued function to be a probability density/mass function, aka, probability function: 1) 2) ∫ for any or ∑ , . Generally, we categorize all real-valued random variables in 3 groups: (1) Discrete Random Variables (its state space is a discrete subset in ); (2) Continuous Random Variables (its state space is a “continuous” subset in ); (3) Partially Discrete and Partially Continuous Random Variables. The Distribution of a Discrete/Continuous Random Variable is called its Probability Mass/Density Function because of a superficial difference in mathematical treatments and graphical representation. Discrete Random Variable Probability Density Function Probability Mass Function Cumulative Distribution Function Expectation {} Continuous - { [ } { } - { ∑ } {} ∫ Problems 1. Suppose that you are invited to play a game with the following rules: One of the numbers 2, … , 12 is chosen at random by throwing a pair of dice and adding the numbers shown. You win 9 dollars in case 2, 3, 11, or 12 comes out, or lose 10 dollars if the outcome is 7. Otherwise, you do not win or lose anything. This defines a function on the set of all possible outcomes {2, … , 12}, the value of the function being the corresponding gain (or loss if the value is negative). What is the probability that the function takes positive values, i.e,, that you will win some money? Solution. As shown in the figure, the answer is . Remark. The Choice of sample space is relative. As long as a set is the domain of some random variable, it is a sample space in effect, although it might also be serving as the state space of another random variable. An example is the state space 1 above. Further in this case, the composite function of random variable 1 and 2 above becomes a new random variable sending elements directly from the sample space to state space 2. Also note that the probability distribution on each of the three spaces above must be “consistent.” This consistency makes it possible to define a random variable between any pairs from the three sets. 2. An urn contains 7 white balls numbered 1, 2, …, 7 and 3 black balls numbered 8, 9, 10. Five balls are randomly selected without replacement. Now give the distribution: I. II. III. IV. of the number of white balls in the sample; of the minimum number in the sample; of the maximum number in the sample; of the minimum number of balls needed for selecting a white ball. Solution. I. As show in the figure (there we use 0 to denote 10 for neatness), the random variable is sending elements from the sample space {12345, 12346, ……, 67890} to the state space {2,3,4,5}. The P.M.F. on the state space is tallied as the following: = 2 ( )( ) ( ) 3 ( )( ) ( ) 4 ( )( ) ( ) 5 ( )( ) ( ) II. Only 1-6 can possibly be the minimum number in any 5 labels drawn. {1,2,…, 6} will be the state space. 1 () ( ) () ( ) 2 () ( ) 3 () ( ) 4 () ( ) 5 () ( ) 6 III. Only 5-10 can possibly be the maximum number in any 5 labels drawn. {5,6,7,8,9,10} will be the state space. This case is symmetric to II. 10 () ( ) () ( ) 9 () ( ) 8 () ( ) 7 () ( ) 6 () ( ) 5 IV. If you draw 4 balls there must be at least 1 white since there are only 3 blacks. If you draw 3 balls it is possible that all the 3 are blacks. Therefore the state space is the set {1,2,3,4}. 1 2 3 4 3. A random variable is called discrete whenever there is a countable set A random variable is said to have the binomial distribution , where {} for Solution. First be aware of the following identity from elementary algebra: () Substitute and by and () () ( ) () such that and [ . , if . Verify that such a random variable is discrete. () respectively will give ∑( ) This means there exists a countable set { discrete random variable. } such that { } . Therefore is a 4. The amount of bread (in hundreds of kilos) that a bakery sells in a day is a random variable with density { a) Find the value of which makes a probability density function. b) What is the probability that the number of kilos of bread that will sold in a day is, (i) more than 300 kilos? (ii) between 150 and 450 kilos? c) Denote by and the events in (i) and (ii), respectively. Are and independent events? Solution. a) First, must be to make ∫ . Second, ∫ has to make ∫ true: b) Since (i) (ii) c) { is continuous at ∫ ∫ } , ∫ , therefore ∫ ∫ . . { { Now that } } , , . ∫ . , they are independent events. 5. A number is randomly chosen from the interval (0,1). What is the probability that: a) its first decimal digit will be a 1; b) its second decimal digit will be a 5; c) the first decimal digit of its square root will be a 3? Solution. First, some clarification. A “number” here means a point of on the axis of real numbers. The usual representation of such a point is called the “decimal expansion” of the number, using digits 0~9. For example, the mid-point of the interval (0,1) has decimal expansion , or equivalently, . a) Since the manner of drawing a point is random, each of the possible first decimal digits will be equiprobable, making as the answer. b) An unconscious reasoning to this part will be “there is no difference between a) and b) except for the superficial difference in positions of the digits”, leading to the same as the answer. A more analytical way of thinking about the two questions is to consider two random variables { { } } to be the (deterministic) mechanisms that tell you the first and second decimal digits, respectively, of the number (randomly) drawn. From the graphs, it is clear the answers derived from our previous unconscious reasoning are correct. c) Define the random variable √ { { } } 6. Find an example of two different random variables Solution. and with the same distribution . Question 1 already provide us such an example: is the random variable 2, and is the composite of random variables 1 and 2. Thus they share the same state space and distribution. Another example is to consider tossing a fair coin with sample space { } R.V. is defined as: 7. (a) A project will bring $1M profit if completed. If the probability of completion is 80%, what is the expected profit? (b) For every set of real numbers, we define the indicator function by { Show that [ Solution. (a) $0.8M. (b) [ { } { } { } ∫ ∫ Or, from the figure we have the equation: [ [ ( { }) {} [ ∫ {} ...
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