Math Problem Statement
Suppose the distribution function of X is given by
F(b) =
⎧⎪⎨
⎪⎩
0, b < 0
12
, 0≤ b < 1
1, 1 ≤ b < ∞
Solution
The given distribution function of a random variable is piecewise and can be described as follows:
\begin{cases} 0, & b < 0 \\ \frac{1}{2}, & 0 \leq b < 1 \\ 1, & 1 \leq b < \infty \end{cases}$$ ### Interpretation: - For $$b < 0$$, the cumulative distribution function (CDF) $$F(b)$$ is 0, meaning the probability that $$X$$ is less than any value $$b < 0$$ is zero. - For $$0 \leq b < 1$$, $$F(b)$$ is constant at $$\frac{1}{2}$$, meaning the probability that $$X$$ is less than $$b$$ (within this range) is $$0.5$$. This suggests that the random variable $$X$$ has a point mass or discontinuity at this interval. - For $$b \geq 1$$, $$F(b) = 1$$, meaning the probability that $$X$$ is less than or equal to 1 is 1. So, after $$b = 1$$, the cumulative probability has reached its maximum. ### Implications: - This function represents a discrete random variable because of the discontinuity and the jumps in the CDF. Specifically, $$X$$ takes values at specific points rather than being continuous across an interval. - $$X$$ could possibly be a **Bernoulli** or another discrete distribution due to the step-like behavior of the function. ### Questions you may want to explore: 1. What kind of random variable is represented by this CDF (discrete, continuous, or mixed)? 2. How would the probability mass function (PMF) of $$X$$ look based on this CDF? 3. What is the expectation (mean) of the random variable $$X$$ given the CDF? 4. Can we find the variance of $$X$$ from the given CDF? 5. How would you interpret the probability that $$X = 1$$? ### Tip: When analyzing a CDF, always observe the points where the function jumps to determine the discrete probabilities and focus on how the function behaves for continuous sections. Would you like a more detailed breakdown on any of these points or another related topic?Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Probability
Cumulative Distribution Function (CDF)
Discrete Random Variable
Formulas
F(b) = { 0, b < 0; 1/2, 0 ≤ b < 1; 1, 1 ≤ b < ∞ }
Theorems
Properties of Cumulative Distribution Functions
Discreteness in Probability Distributions
Suitable Grade Level
Undergraduate (Probability Theory)