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Concept explainers
Let Y be a random variable with
a Find the density function of U1 = 2Y − 1.
b Find the density function of U2 = 1 − 2Y.
c Find the density function of U3 = Y2.
d Find E(U1), E(U2), and E(U3) by using the derived density
e Find E(U1), E(U2), and E(U3) by the methods of Chapter 4.
a.
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Find the density function for
Answer to Problem 1E
The density function for
Explanation of Solution
Calculation:
From the given information, the probability density function for Y is
The distribution function for Y is,
From the given information, the random variable
Consider the distribution function for
Limits for the random variable U1:
The range for the random variable Y is from 0 to 1 and
For Y= 0, the value of U1 is ̶ 1.
For
Hence, the range for the random variable U1 is from ̶ 1 to 1.
The probability density function for
b.
![Check Mark](/static/check-mark.png)
Find the density function for
Answer to Problem 1E
The density function for
Explanation of Solution
Calculation:
From the given information,
Consider the distribution function for
Limits for the random variable U2:
The range for the random variable Y is from 0 to 1 and
For Y= 0, the value of U2 is 1.
For
Hence, the range for the random variable U2 is from ̶ 1 to 1.
The probability density function for
c.
![Check Mark](/static/check-mark.png)
Find the density function for
Answer to Problem 1E
The density function for
Explanation of Solution
Calculation:
From the given information,
Consider the distribution function for
The probability density function for
d.
![Check Mark](/static/check-mark.png)
Find the value of
Find the value of
Find the value of
Answer to Problem 1E
The value of
The value of
The value of
Explanation of Solution
Calculation:
The density function for
Consider,
The density function for
Consider,
The density function for
Consider,
e.
![Check Mark](/static/check-mark.png)
Find the value of
Find the value of
Find the value of
Answer to Problem 1E
The value of
The value of
The value of
Explanation of Solution
Calculation:
Result:
Let X be the random variable, then
The density function for Y is
Consider,
Consider,
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Chapter 6 Solutions
Mathematical Statistics with Applications
- Let X and Y be independent random variables each having density function 2r fx (x) = 2 e * for x20 elsewhere Find: E (X+ Y)arrow_forwardA random variable, Y, has probability density function: -1< y<0 f(y) c+y 0 < y <1 otherwise a) Find the value of the constant, c. b) Find E(Y). c) Find Var(Y).arrow_forwardLet x be a continuous random variable with the density function: f(x) = 3e-3x when x>0 and 0 else Find the variance of the random variable x.arrow_forward
- Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning
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