For a Standard Normal Distribution, Which of the Following Expressions Always Equals 1?
In statistics, the standard normal distribution plays a vital role in determining probabilities of events. A standard normal distribution has a mean of 0 and a standard deviation of 1. When working with this distribution, certain expressions will always equal 1. Let’s explore some of these expressions and understand why they hold true.
1. The probability density function (PDF): The area under the curve of a probability density function is always equal to 1. This means that the integral of the PDF from negative infinity to infinity is 1.
2. The cumulative distribution function (CDF): The CDF of a standard normal distribution gives the probability that a random variable is less than or equal to a specific value. The CDF at negative infinity is always 0, while the CDF at positive infinity is always 1.
3. The sum of probabilities: Since a standard normal distribution encompasses the entire range of possible values, the probability of obtaining any value within this range is 1.
4. The expected value: The expected value of a random variable following a standard normal distribution is always equal to its mean, which is 0.
5. The Z-score: The Z-score represents the number of standard deviations a given value is away from the mean. The probability of obtaining a Z-score of 0 is always 1, as it represents the mean itself.
6. The complement rule: The probability of an event and its complement always add up to 1. In a standard normal distribution, the complement of a certain probability is the probability of obtaining a value greater than or equal to that value.
7. The standard normal table: The values in a standard normal table are calculated in such a way that the cumulative probabilities in the table sum up to 1.
Q1. What is the standard normal distribution?
A1. The standard normal distribution is a specific type of probability distribution with a mean of 0 and a standard deviation of 1.
Q2. Why is the area under the curve of the PDF equal to 1?
A2. The area under the curve represents the total probability of all possible outcomes, which must sum up to 1.
Q3. Why is the CDF at positive infinity equal to 1?
A3. Since the standard normal distribution encompasses all possible values, the probability of obtaining a value greater than any specific value is 1.
Q4. Why is the expected value 0?
A4. The expected value represents the mean of the distribution, which is 0 for a standard normal distribution.
Q5. What does a Z-score of 0 indicate?
A5. A Z-score of 0 indicates that the value is equal to the mean of the distribution.
Q6. Can the probability of an event be greater than 1 in a standard normal distribution?
A6. No, the probability of any event in a standard normal distribution cannot exceed 1.
Q7. Why do the cumulative probabilities in a standard normal table add up to 1?
A7. The standard normal table is constructed in a way that ensures the cumulative probabilities sum up to 1, providing a comprehensive reference for standard normal distribution calculations.
Understanding these expressions and their properties is crucial for statistical analysis and inference when dealing with the standard normal distribution.