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

Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.1: Continuous Probability Models
Problem 25E
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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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