Total marks 17 4. Let (,,P) be a probability space and let X : → R be a ran- dom variable that has Gamma(2, 1) distribution, i.e., the distribution of the random variable X is the probability measure on ((0, ∞), B((0, ∞))) given by (i) dPx(x) = xex dx. Find the characteristic function of the random variable X. [8 Marks] (ii) Using the result of (i), calculate the first three moments of the random variable X, i.e., E(X") for n = 1, 2, 3. Using Markov's inequality involving E(X³), (iii) probability P(X > 10). [6 Marks] estimate the [3 Marks]
Total marks 17 4. Let (,,P) be a probability space and let X : → R be a ran- dom variable that has Gamma(2, 1) distribution, i.e., the distribution of the random variable X is the probability measure on ((0, ∞), B((0, ∞))) given by (i) dPx(x) = xex dx. Find the characteristic function of the random variable X. [8 Marks] (ii) Using the result of (i), calculate the first three moments of the random variable X, i.e., E(X") for n = 1, 2, 3. Using Markov's inequality involving E(X³), (iii) probability P(X > 10). [6 Marks] estimate the [3 Marks]
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 22E
Related questions
Question
![Total marks 17
4.
Let (,,P) be a probability space and let X : → R be a ran-
dom variable that has Gamma(2, 1) distribution, i.e., the distribution of
the random variable X is the probability measure on ((0, ∞), B((0, ∞)))
given by
(i)
dPx(x) = xex dx.
Find the characteristic function of the random variable X.
[8 Marks]
(ii) Using the result of (i), calculate the first three moments of
the random variable X, i.e., E(X") for n = 1, 2, 3.
Using Markov's inequality involving E(X³),
(iii)
probability P(X > 10).
[6 Marks]
estimate the
[3 Marks]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd66c7573-6777-48ff-9bfb-9b3df1a769a6%2Ff59a54c1-68fb-40bd-9146-30145c4ccbbe%2Fc5qwtlq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Total marks 17
4.
Let (,,P) be a probability space and let X : → R be a ran-
dom variable that has Gamma(2, 1) distribution, i.e., the distribution of
the random variable X is the probability measure on ((0, ∞), B((0, ∞)))
given by
(i)
dPx(x) = xex dx.
Find the characteristic function of the random variable X.
[8 Marks]
(ii) Using the result of (i), calculate the first three moments of
the random variable X, i.e., E(X") for n = 1, 2, 3.
Using Markov's inequality involving E(X³),
(iii)
probability P(X > 10).
[6 Marks]
estimate the
[3 Marks]
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