Math Problem Statement
. A discrete random variable Y has a pmf given by ( ) for 0,1,2,..... 4 3 P Y y c y x Find the value of the constant c , PX 3 and PX 3
Solution
Problem Breakdown:
The problem states that is a discrete random variable with the probability mass function (pmf):
Additionally, the probabilities must satisfy the property of a probability mass function:
We will:
- Solve for the constant using the above condition.
- Compute and .
Step 1: Solve for
From the pmf, we write:
However, this series is infinite. It is common to cap the range of to some maximum value (e.g., 4 here, ). Thus, the summation becomes:
Calculating the sum:
So,
Step 2: Compute
For , the possible values of are and . Thus:
Using , we calculate:
So:
Step 3: Compute
For , the possible values of are and . Thus:
Using , we calculate:
So:
Final Answers:
- ,
- ,
- .
Questions:
- How does a probability mass function (pmf) differ from a probability density function (pdf)?
- Can this approach extend to cases where has an infinite range?
- What properties of ensure that the total probability is always 1?
- How would you compute (expected value) for this pmf?
- If , how would the solution for change?
Tip:
Always verify that the sum of probabilities in a pmf equals 1, as this ensures it is a valid probability distribution.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Probability
Discrete Random Variables
Probability Mass Function (PMF)
Formulas
Sum of probabilities in a PMF: ΣP(Y = y) = 1
Probability computations: P(A) = ΣP(Y = y) for y in A
Theorems
Normalization property of PMF: The sum of all probabilities must equal 1.
Suitable Grade Level
Grades 11-12
Related Recommendation
Probability Distribution: Calculate P(1 ≤ Y ≤ 4) and the Most Likely Value of Y
Find the Probability Distribution and CDF of a Random Variable X with Given Relationships
Calculating Probabilities Using a Discrete Probability Mass Function
Calculate P(X ≤ 3) for Discrete Probability Distribution
Finding k and Probabilities for a Discrete Random Variable X