Answer :
Let's analyze the expressions one by one through step-by-step reasoning:
### Expression 1:
[tex]\[ P(z \leq -a) - P(-a \leq z \leq a) - P(z \geq a) \][/tex]
- Part 1: \( P(z \leq -a) \)
This represents the probability that the standard normal variable \( z \) is less than or equal to \(-a\).
- Part 2: \( P(-a \leq z \leq a) \)
This represents the probability that the standard normal variable \( z \) is between \(-a\) and \( a \).
- Part 3: \( P(z \geq a) \)
This represents the probability that the standard normal variable \( z \) is greater than or equal to \( a \).
### Expression 2:
[tex]\[ P(z \leq -a) - P(-a \leq z \leq a) + P(z \geq a) \][/tex]
### Expression 3:
[tex]\[ P(z \leq -a) + P(-a \leq z \leq a) - P(z \geq a) \][/tex]
### Expression 4:
[tex]\[ P(z \leq -a) + P(-a \leq z \leq a) + P(z \geq a) \][/tex]
#### Step-by-Step Breakdown:
1. Standard Normal Distribution Property:
The total probability for a standard normal distribution is always equal to 1.
2. Key Relationships:
- \( P(z \leq -a) + P(-a \leq z \leq a) + P(z \geq a) = 1 \)
- \( P(z \leq -a) + P(z \geq a) \) covers the entire distribution as \( z \) cannot be in both ranges simultaneously.
- \( P(-a \leq z \leq a) \) is the probability within the bounds \(-a\) to \(a\).
When we analyze these expressions, we notice that:
- \( P(z \leq -a) \) covers the lower tail.
- \( P(z \geq a) \) covers the upper tail.
- \( P(-a \leq z \leq a) \) covers the central interval.
- The sum of probabilities over these mutually exclusive events should be:
[tex]\[ P(z \leq -a) + P(-a \leq z \leq a) + P(z \geq a) = 1 \][/tex]
Hence, by looking at our given options, we need to find which expression sums up correctly to 1:
- Option 1: \( P(z \leq -a) - P(-a \leq z \leq a) - P(z \geq a) \) simplifies incorrectly and doesn't sum to 1.
- Option 2: \( P(z \leq -a) - P(-a \leq z \leq a) + P(z \geq a) \) simplifies correctly and equals to 1.
- Option 3: \( P(z \leq -a) + P(-a \leq z \leq a) - P(z \geq a) \) simplifies incorrectly and doesn't sum to 1.
- Option 4: \( P(z \leq -a) + P(-a \leq z \leq a) + P(z \geq a) \) results in exceeding 1.
Therefore, the expression that must always be equal to 1 for a standard normal distribution is:
[tex]\[ P(z \leq -a) - P(-a \leq z \leq a) + P(z \geq a) \][/tex]
Thus, the correct answer is option 2.
### Expression 1:
[tex]\[ P(z \leq -a) - P(-a \leq z \leq a) - P(z \geq a) \][/tex]
- Part 1: \( P(z \leq -a) \)
This represents the probability that the standard normal variable \( z \) is less than or equal to \(-a\).
- Part 2: \( P(-a \leq z \leq a) \)
This represents the probability that the standard normal variable \( z \) is between \(-a\) and \( a \).
- Part 3: \( P(z \geq a) \)
This represents the probability that the standard normal variable \( z \) is greater than or equal to \( a \).
### Expression 2:
[tex]\[ P(z \leq -a) - P(-a \leq z \leq a) + P(z \geq a) \][/tex]
### Expression 3:
[tex]\[ P(z \leq -a) + P(-a \leq z \leq a) - P(z \geq a) \][/tex]
### Expression 4:
[tex]\[ P(z \leq -a) + P(-a \leq z \leq a) + P(z \geq a) \][/tex]
#### Step-by-Step Breakdown:
1. Standard Normal Distribution Property:
The total probability for a standard normal distribution is always equal to 1.
2. Key Relationships:
- \( P(z \leq -a) + P(-a \leq z \leq a) + P(z \geq a) = 1 \)
- \( P(z \leq -a) + P(z \geq a) \) covers the entire distribution as \( z \) cannot be in both ranges simultaneously.
- \( P(-a \leq z \leq a) \) is the probability within the bounds \(-a\) to \(a\).
When we analyze these expressions, we notice that:
- \( P(z \leq -a) \) covers the lower tail.
- \( P(z \geq a) \) covers the upper tail.
- \( P(-a \leq z \leq a) \) covers the central interval.
- The sum of probabilities over these mutually exclusive events should be:
[tex]\[ P(z \leq -a) + P(-a \leq z \leq a) + P(z \geq a) = 1 \][/tex]
Hence, by looking at our given options, we need to find which expression sums up correctly to 1:
- Option 1: \( P(z \leq -a) - P(-a \leq z \leq a) - P(z \geq a) \) simplifies incorrectly and doesn't sum to 1.
- Option 2: \( P(z \leq -a) - P(-a \leq z \leq a) + P(z \geq a) \) simplifies correctly and equals to 1.
- Option 3: \( P(z \leq -a) + P(-a \leq z \leq a) - P(z \geq a) \) simplifies incorrectly and doesn't sum to 1.
- Option 4: \( P(z \leq -a) + P(-a \leq z \leq a) + P(z \geq a) \) results in exceeding 1.
Therefore, the expression that must always be equal to 1 for a standard normal distribution is:
[tex]\[ P(z \leq -a) - P(-a \leq z \leq a) + P(z \geq a) \][/tex]
Thus, the correct answer is option 2.
