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Questions and Answers
What defines a random variable?
In the example of tossing a fair coin multiple times, what does the random variable X represent?
What is the correct definition of a cumulative distribution function (CDF) for a random variable X?
For the cumulative distribution function FX(x), what is the value when x is less than 0?
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What property ensures that the cumulative distribution function (CDF) has only jump discontinuities?
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Which limit property must hold true for a CDF as x approaches infinity?
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If FX(x) = 1 - e^{-λx} for x ≥ 0, what type of function is FX(x) being described?
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What does the limit lim FX(x - h) = FX(x) - P(X = x) represent?
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How is a function F confirmed to be a cumulative distribution function based on properties (1)-(3)?
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What impact does a CDF have at points of discontinuity?
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For a function to remain a CDF while demonstrating non-decreasing behavior, what must its slope not do?
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When evaluating the limit of a CDF as x approaches negative infinity, what is expected?
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Study Notes
Random Variables
- A random variable is defined as a function X : S → R.
- Example of X: Tossing a fair coin n times, where X counts the number of heads.
- Another example: Throwing a fair die twice, where X is the sum of the outcomes.
- A reliability test of a battery can define X such that X1(ω) = ω and X2 = 1(2,∞) if focusing on battery life beyond 2 years.
Probability Distributions
- Distributions provide probabilities for specific outcomes.
- Example probabilities for a scenario with n=2 are:
- P(X = 0) = 1/4, P(X = 1) = 1/2, P(X = 2) = 1/4.
- Another example outlines probabilities for outcomes from rolling a die multiple times, such as:
- P(X = 2) = 1/36, P(X = 3) = 2/36, up to P(X = 12) = 1/36.
Cumulative Distribution Function (CDF)
- The CDF of a random variable X is defined as FX(x) = P(X ≤ x), mapping R → [0, ∞).
- CDF examples:
- For a variable with defined ranges:
- FX(x) = 0 if x < 0,
- FX(x) = 1/4 if 0 ≤ x < 1,
- FX(x) = 1 if x ≥ 2.
- For a variable with defined ranges:
- Another variable's CDF example includes probabilities calculated for ranges, culminating in FX(x) = 1 if x ≥ 12.
Properties of Cumulative Distribution Function
- The following properties characterize a CDF:
- FX(·) is non-decreasing, exhibiting only jump discontinuities.
- As x approaches infinity, the limit of FX(x) approaches 1; as x approaches negative infinity, it approaches 0.
- CDFs are right-continuous: lim FX(x + h) = FX(x) for all x ∈ R.
- A left-hand limit property: lim FX(x - h) = FX(x) - P(X = x) for all x ∈ R.
Theorem on CDF
- A function F meeting properties (1)-(3) qualifies as a CDF, reinforcing the requirements for valid cumulative distribution functions.
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Description
Explore the concept of random variables through practical examples in this lecture of Probability Theory (MA225). Learn how random variables can be defined through functions and illustrated with various scenarios like coin tosses and dice throws. This quiz will test your understanding of these foundational elements in probability.
