Math Problem Statement
Example 3.2: Let (X, Y ) be a DRV with JPMF fX,Y (x, y) = ( cy if x = 1, 2, . . . , n; y = 1, 2, . . . , n; x ≤ y 0 otherwise.
Solution
To solve Example 3.2, we need to analyze the Joint Probability Mass Function (JPMF) given for the discrete random variables and .
Step 1: Understanding the JPMF
The joint probability mass function is given as:
\begin{cases} cy & \text{if } x = 1, 2, \dots, n; y = 1, 2, \dots, n; \text{ and } x \leq y \\ 0 & \text{otherwise} \end{cases}$$ ### Step 2: Determine the Constant $$c$$ The first step in dealing with a joint PMF like this one is to determine the constant $$c$$. We know that the sum of all joint probabilities must equal 1. Therefore, we sum over all valid pairs $$(x, y)$$ and set the result equal to 1. $$\sum_{x=1}^n \sum_{y=1}^n f_{X,Y}(x, y) = 1$$ Substituting the given function $$f_{X,Y}(x, y) = cy$$ where $$x \leq y$$: $$\sum_{x=1}^n \sum_{y=x}^n cy = 1$$ Now, the sum is taken over $$y$$ starting from $$x$$ to $$n$$ because $$x \leq y$$. So, we can write: $$c \sum_{x=1}^n \sum_{y=x}^n y = 1$$ Let's calculate the inner sum: $$\sum_{y=x}^n y = \frac{n(n+1)}{2} - \frac{(x-1)x}{2}$$ This simplifies to: $$\sum_{y=x}^n y = \frac{n(n+1) - (x-1)x}{2} = \frac{(n-x+1)(n+x)}{2}$$ Now the equation becomes: $$c \sum_{x=1}^n \frac{(n-x+1)(n+x)}{2} = 1$$ To simplify further, calculate the sum $$\sum_{x=1}^n (n-x+1)(n+x)$$ and then solve for $$c$$. ### Step 3: Solve for $$c$$ Simplify the double sum, solve for $$c$$, and you'll obtain the constant that normalizes the probability mass function. ### Step 4: Final JPMF Once you have $$c$$, you can express the final joint PMF $$f_{X,Y}(x, y)$$. --- Would you like to go through the detailed calculation for finding the value of $$c$$ or explore related questions? ### Related Questions: 1. What is the marginal PMF of $$X$$ from the given joint PMF? 2. What is the marginal PMF of $$Y$$ from the given joint PMF? 3. How would you calculate the conditional PMF $$f_{X|Y}(x|y)$$? 4. What is the expectation $$E[X]$$ given this joint PMF? 5. How would the joint PMF change if the condition $$x \leq y$$ were altered? **Tip:** When dealing with joint distributions, always ensure the sum of probabilities equals 1.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability Theory
Joint Probability Mass Function (JPMF)
Discrete Random Variables
Formulas
f_{X,Y}(x, y) = cy if x ≤ y
∑_{x=1}^n ∑_{y=x}^n cy = 1
Theorems
Law of Total Probability
Normalization Condition of PMF
Suitable Grade Level
University level (Probability and Statistics)
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