
A First Course in Probability (10th Edition)
10th Edition
ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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![Consider a continuous random variable X with the following PDF:
ſc −1≤x≤ 0 or 1 ≤ x ≤ 3
- {%
otherwise
(a) Sketch the PDF of X. Be sure to label your axes.
(b) Determine the value of c that satisfies the normalization property. Set c to this value for
the remainder of the problem. This can be done without integration, so your answer should
be a number.
fx(x)=
(c) What is the expected value of X?
(d) What is the variance of X?
(e) What is E[2X² - 3X + 1]?
(f) Calculate the probability that |X| is less than 2. (Your answer can be an integral, but you
can also take advantage of the simple structure of the PDF.)
(g) Let B
=
(-2,2). Determine the conditional PDF fx|B(x) of X given the event {X € B}.
(h) Calculate the probability that X < 0 given that X is less than 2.
(i) Calculate the conditional expectation E[X|B] of X given that |X| is less than 2.](https://content.bartleby.com/qna-images/question/02681530-b6ca-4d72-bfd1-7260256101ab/ffef05df-c6ca-475b-8740-aa508ff5ad99/0t0dihs_thumbnail.png)
Transcribed Image Text:Consider a continuous random variable X with the following PDF:
ſc −1≤x≤ 0 or 1 ≤ x ≤ 3
- {%
otherwise
(a) Sketch the PDF of X. Be sure to label your axes.
(b) Determine the value of c that satisfies the normalization property. Set c to this value for
the remainder of the problem. This can be done without integration, so your answer should
be a number.
fx(x)=
(c) What is the expected value of X?
(d) What is the variance of X?
(e) What is E[2X² - 3X + 1]?
(f) Calculate the probability that |X| is less than 2. (Your answer can be an integral, but you
can also take advantage of the simple structure of the PDF.)
(g) Let B
=
(-2,2). Determine the conditional PDF fx|B(x) of X given the event {X € B}.
(h) Calculate the probability that X < 0 given that X is less than 2.
(i) Calculate the conditional expectation E[X|B] of X given that |X| is less than 2.
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Can you please do (f), (g), (h), and (i)?
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Can you please do (f), (g), (h), and (i)?
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