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2. A random variable X has a probability density function (pdf) given by , 0< x < 1, f(x) = c, 1

2. A random variable X has a probability density function (pdf) given by , 0< x < 1, f(x) = c, 1

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 31E
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2. A random variable X has a probability density function (pdf) given by , 0< x < 1, 2 f(x) = C, 1<x < 3, 0, elsewhere, for some real constant c. (a) Determine the value of the constant c and sketch the graph of f(x). In the following questions, replace c by its numerical value computed in (a). (b) Find the cumulative distribution function (cdf) F(x) of X, and sketch its graph. (c) Find P(X < ). (d) Find P( < X < ). (e) What is P(X = 1)? %| (f) Find the moment-generating function (mgf) Mx(t) of X. Justify why it is well defined for any real finite value of t. (f) Find the moment-generating function (mgf) Mx(t) of X. Justify why it is well defined for any real finite value of t. (g) Find the 75th percentile of the distribution. Namely, find the value of T0.75 SO that P(X < Τ0.75)F(T0.T3)0.75. (h) Find the conditional probability P(X > |X > ).
Transcribed Image Text:2. A random variable X has a probability density function (pdf) given by , 0< x < 1, 2 f(x) = C, 1<x < 3, 0, elsewhere, for some real constant c. (a) Determine the value of the constant c and sketch the graph of f(x). In the following questions, replace c by its numerical value computed in (a). (b) Find the cumulative distribution function (cdf) F(x) of X, and sketch its graph. (c) Find P(X < ). (d) Find P( < X < ). (e) What is P(X = 1)? %| (f) Find the moment-generating function (mgf) Mx(t) of X. Justify why it is well defined for any real finite value of t. (f) Find the moment-generating function (mgf) Mx(t) of X. Justify why it is well defined for any real finite value of t. (g) Find the 75th percentile of the distribution. Namely, find the value of T0.75 SO that P(X < Τ0.75)F(T0.T3)0.75. (h) Find the conditional probability P(X > |X > ).
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