Math Problem Statement
Let X and Y be continuous random variables with joint density function
fx,y(x, y)=3x/ 4 for 0 < x < 2 and 0<y<2-x
What is P(X > 1)?
Solution
To find the probability for the continuous random variables and with the given joint density function, we need to integrate the joint density over the region where .
Step-by-Step Solution:
-
Understand the Joint Density Function:
The joint density function is given by:
-
Set up the Probability :
We need to integrate the joint density function over the region where . Since ranges from 1 to 2, and for each value of , ranges from 0 to , we have:
-
Substitute the Joint Density Function:
Substituting , the probability becomes:
-
Integrate with Respect to First:
The inner integral with respect to is:
So, the probability becomes:
-
Simplify and Integrate with Respect to :
Distribute :
Separate the integrals:
Compute each integral:
Plugging in the limits from 1 to 2:
[ P(X > 1) = \frac{3}{2} \left[ \frac{x^2}{2} \right]{1}^{2} - \frac{3}{4} \left[ \frac{x^3}{3} \right]{1}^{2}. ]
Evaluate the limits:
Therefore:
Simplify the terms:
Final Answer:
Would you like any further details or have any questions?
Relative Questions:
- How do you find the marginal density function of from a joint density function?
- What is the meaning of a joint density function in probability theory?
- How do you compute the probability for this problem?
- How do you verify if a joint density function is valid?
- What is the difference between continuous and discrete random variables?
Tip: To find a marginal distribution from a joint density, integrate the joint density over all possible values of the other variable.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Probability Theory
Joint Density Function
Integral Calculus
Formulas
Joint density function
Integral of a continuous function
Theorems
-
Suitable Grade Level
Advanced Mathematics
Related Recommendation
Conditional Probability P(Y > 2 | X > 1) Calculation
Probability Calculation for Joint Random Variables with Conditions on x and y
Calculate P(2 < X < 5) for a Continuous Random Variable with Density f(x) = 1/(1+x)^2
Joint Density Function and Independence of X and Y: Probability Solution
Calculate Probabilities Using CDF for a Continuous Random Variable