Step-by-step explanation:
To show that \( F(x) \), a cumulative distribution function (CDF), lies between 0 and 1 for all \( x \), we will utilize the properties of CDFs:
1. **Definition of Cumulative Distribution Function (CDF):**
\( F(x) \) is defined as \( F(x) = P(X \leq x) \), where \( X \) is a random variable. By definition, \( F(x) \) gives the probability that \( X \) takes on a value less than or equal to \( x \).
2. **Properties of CDF:**
- \( F(x) \geq 0 \) for all \( x \): This is because probabilities are non-negative.
- \( F(x) \leq 1 \) for all \( x \): This is because the probability of any event cannot exceed 1.
Let's formally prove these properties:
### Proof:
**1. \( F(x) \geq 0 \) for all \( x \):**
Since \( F(x) = P(X \leq x) \), and probabilities are non-negative, we have:
\[ F(x) = P(X \leq x) \geq 0 \]
Therefore, \( F(x) \geq 0 \) for all \( x \).
**2. \( F(x) \leq 1 \) for all \( x \):**
Again, since \( F(x) = P(X \leq x) \), and \( P(X \leq x) \) represents the cumulative probability up to \( x \), it cannot exceed 1:
\[ F(x) = P(X \leq x) \leq 1 \]
Therefore, \( F(x) \leq 1 \) for all \( x \).
### Conclusion:
From the above two properties, we conclude that \( F(x) \) indeed lies between 0 and 1 for any \( x \):
\[ 0 \leq F(x) \leq 1 \]
This completes the proof that \( F(x) \), being a cumulative distribution function (CDF), satisfies the condition of lying between 0 and 1 for all \( x \).
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