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Assignments - 1 Let X be a discrete random variable that...

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1) Let X be a discrete random variable that attains values 1, 2 and 5 with probability 1/8, 1/4 and 5/8 respectively. Find: a) E(X); b) Var(X); c) E(2X+3) d) The Standard deviation of 2X+3 SOLUTION: x 1 2 5 P(X=x) 1/8 1/4 5/8 (a) E(X) = Ʃ x=1,2,5 x P(X=x) = 1*(1/8) + 2*(1/4) +5*(5/8) = 15/4=3.75 (b) Method 1: Var(X)=E(X 2 )-[E(X)] 2 E(X 2 ) = Ʃ x=1,2,5 x 2 P(X=x) =1 2 *(1/8)+2 2 *(1/4)+5 2 *(5/8) =16.75 Therefore, Var(X)=16.75-(3.75) 2 = 2.6875 Method 2:
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By definition, Var(X) = E[(X-E(X)) 2 ] Var(X)= Ʃ x=1,2,5 (x-E(X)) 2 P(X=x) = Ʃ x=1,2,5 (x-3.75) 2 P(X=x) = (1-3.75) 2 *(1/8)+(2-3.75) 2 *(1/4)+(5-3.75) 2 *(5/8) = 2.6875 (c) By linearity property of expectations, E(2X+3)=2E(X)+3=2(3.75)+3=10.5 (d) Standard deviation of (2X+3)=√Var(2X+3) Recall that Var(aX+b)=a 2 Var(X) where ‘a’ and ‘b’ are constants Therefore Var(2X+3)=2 2 * Var(X)=4*2.6875=10.75 Hence, we have standard deviation of (2X+3)=√10.75≈3.279 2). In order to evaluate the effect of technology in the classroom students in a section of STAT 231 form part of an experiment. The technology involved involves broadcasting the lecture live to another classroom. The lecture is also recorded and can be downloaded for review by any student. Professors at Waterloo want to evaluate this video-conferencing technology to see if it can be used for any course at Waterloo in the future. Evaluation of this section is therefore an experiment, which can help evaluate the technology. It is important to discover if there is a difference in the learning experience between students who attend the lecture in the room with the instructor and those students in the room that receives the broadcast. If there is no difference between the two groups of students then the technology will be used for other courses at Waterloo. The first problem is therefore to discover if there are any differences between the two groups of students. A second related problem is to discover if the ability of students to download lectures for review enhances the learning experience. In order to investigate the first question, during a particular lecture the students in Section 1 are randomly assigned to Room A and Room B. In Room A the lecture is ‘live’, while in B
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it is broadcast. At the end of the lecture the two groups of students are asked to fill-in a questionnaire designed to evaluate their experience. One of the questions was: 1) Do you agree with the following statement. ‘The lecture was highly informative’? 1= Strongly disagree, 2= Mildly disagree, 3= No opinion, 4=Mildly agree, 5= Strongly agree There were 53 students in Room A and 39 in Room B. The number of students in Room A who answered 5 was 19, while the number in Room B the number was 0.
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