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For a standard normal distribution, which of the following expressions must always be equal to 1?

A. [tex] P(z \leq -a) - P(-a \leq z \leq a) - P(z \geq a) [/tex]
B. [tex] P(z \leq -a) - P(-a \leq z \leq a) + P(z \geq a) [/tex]
C. [tex] P(z \leq -a) + P(-a \leq z \leq a) - P(z \geq a) [/tex]
D. [tex] P(z \leq -a) + P(-a \leq z \leq a) + P(z \geq a) [/tex]

Sagot :

GinnyAnswer
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.
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