Let X be a continuous random variable with pdf f(x) = сх 2 C, 0 ≤ x < 2, 2 ≤ x ≤ 3, for some real constant c, and f(x) = 0 elsewhere. (a) Show that c = 1/2, and sketch the graph of f(x). (b) Find the cdf of X and sketch its graph. (c) Find E(X), the expectation of X. (d) Find E(X²), the second moment of X. Then find the variance o² of X. (e) Find M(t) = E(e¹X), the mgf of X. (f) Calculate E(2³).
Let X be a continuous random variable with pdf f(x) = сх 2 C, 0 ≤ x < 2, 2 ≤ x ≤ 3, for some real constant c, and f(x) = 0 elsewhere. (a) Show that c = 1/2, and sketch the graph of f(x). (b) Find the cdf of X and sketch its graph. (c) Find E(X), the expectation of X. (d) Find E(X²), the second moment of X. Then find the variance o² of X. (e) Find M(t) = E(e¹X), the mgf of X. (f) Calculate E(2³).
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text:Let X be a continuous random variable with pdf
f(x)
=
сх
2
C,
0 ≤ x < 2,
2 ≤ x ≤ 3,
for some real constant c, and f(x) = 0 elsewhere.
(a) Show that c = 1/2, and sketch the graph of f(x).
(b) Find the cdf of X and sketch its graph.
(c) Find E(X), the expectation of X.
(d) Find E(X²), the second moment of X. Then find the variance o² of X.
(e) Find M(t) = E(e¹X), the mgf of X.
(f) Calculate E(2³).
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