2. The continuous random variables x and y have known joint probability density function f(x,y) given by x Osysxsl fxy(x,y) = y +1 0 otherwise where is a positive constant. a) Determine the value of the constant c that makes the above a valid joint probability density function. b) Using the result of part a), obtain now the marginal probability density function fx(x) of the random variable x. c) Using the result of part a), obtain now the marginal probability density function f(x) of the random variable y. d) Verify the results of parts b) and c) to make sure that the two probability density functions are valid. e) Are the two random variables statistically independent? f) Obtain the expected value E(XY).
2. The continuous random variables x and y have known joint probability density function f(x,y) given by x Osysxsl fxy(x,y) = y +1 0 otherwise where is a positive constant. a) Determine the value of the constant c that makes the above a valid joint probability density function. b) Using the result of part a), obtain now the marginal probability density function fx(x) of the random variable x. c) Using the result of part a), obtain now the marginal probability density function f(x) of the random variable y. d) Verify the results of parts b) and c) to make sure that the two probability density functions are valid. e) Are the two random variables statistically independent? f) Obtain the expected value E(XY).
Transcribed Image Text:### Joint Probability Density Function of Continuous Random Variables
The continuous random variables \( X \) and \( Y \) have a known joint probability density function \( f_{XY}(x, y) \) given by:
\[
f_{XY}(x, y) =
\begin{cases}
\frac{c \cdot x}{y + 1} & \text{if } 0 \leq y \leq x \leq 1 \\
0 & \text{otherwise}
\end{cases}
\]
where \( c \) is a positive constant.
#### Tasks:
a) **Determine the value of the constant \( c \):**
Find \( c \) such that the function becomes a valid joint probability density function.
b) **Marginal Probability Density Function \( f_X(x) \):**
Using the result from part (a), obtain the marginal probability density function of the random variable \( X \).
c) **Marginal Probability Density Function \( f_Y(y) \):**
Using the result from part (a), obtain the marginal probability density function of the random variable \( Y \).
d) **Verification of Functions:**
Verify the results of parts (b) and (c) to ensure that the two probability density functions are valid.
e) **Statistical Independence:**
Determine whether the two random variables are statistically independent.
f) **Expected Value \( E\{XY\} \):**
Obtain the expected value of the product \( XY \).
Expert Solution
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Step 1: Continuous probability distribution
As per the guidelines of the Bartleby, solution of only first 3 parts are given.
![### Joint Probability Density Function of Continuous Random Variables
The continuous random variables \( X \) and \( Y \) have a known joint probability density function \( f_{XY}(x, y) \) given by:
\[
f_{XY}(x, y) =
\begin{cases}
\frac{c \cdot x}{y + 1} & \text{if } 0 \leq y \leq x \leq 1 \\
0 & \text{otherwise}
\end{cases}
\]
where \( c \) is a positive constant.
#### Tasks:
a) **Determine the value of the constant \( c \):**
Find \( c \) such that the function becomes a valid joint probability density function.
b) **Marginal Probability Density Function \( f_X(x) \):**
Using the result from part (a), obtain the marginal probability density function of the random variable \( X \).
c) **Marginal Probability Density Function \( f_Y(y) \):**
Using the result from part (a), obtain the marginal probability density function of the random variable \( Y \).
d) **Verification of Functions:**
Verify the results of parts (b) and (c) to ensure that the two probability density functions are valid.
e) **Statistical Independence:**
Determine whether the two random variables are statistically independent.
f) **Expected Value \( E\{XY\} \):**
Obtain the expected value of the product \( XY \).](https://content.bartleby.com/qna-images/question/9328926a-b343-4d47-8a00-cbd9e6e36ac2/afd82448-f5db-488f-82fa-4da35c6511b1/5einod6_processed.jpeg)
