Question
Let X be a random variable with the standard normal distribution. Find: P(X > 1.23)
Answers
GPT-4o mini
To find \( P(X > 1.23) \) for a random variable \( X \) that follows a standard normal distribution, we can use the cumulative distribution function (CDF) of the standard normal distribution, which is denoted as \( \Phi(z) \).
The probability we want can be expressed in terms of the CDF as follows:
\[
P(X > 1.23) = 1 - P(X \leq 1.23) = 1 - \Phi(1.23)
\]
Next, we look up the value of \( \Phi(1.23) \) in the standard normal distribution table or use a calculator or statistical software that provides the CDF for the standard normal distribution.
From tables or calculator, we find:
\[
\Phi(1.23) \approx 0.8907
\]
Thus, the probability \( P(X > 1.23) \) can be calculated as:
\[
P(X > 1.23) = 1 - 0.8907 \approx 0.1093
\]
Therefore, the final answer is:
\[
P(X > 1.23) \approx 0.1093
\]
The probability we want can be expressed in terms of the CDF as follows:
\[
P(X > 1.23) = 1 - P(X \leq 1.23) = 1 - \Phi(1.23)
\]
Next, we look up the value of \( \Phi(1.23) \) in the standard normal distribution table or use a calculator or statistical software that provides the CDF for the standard normal distribution.
From tables or calculator, we find:
\[
\Phi(1.23) \approx 0.8907
\]
Thus, the probability \( P(X > 1.23) \) can be calculated as:
\[
P(X > 1.23) = 1 - 0.8907 \approx 0.1093
\]
Therefore, the final answer is:
\[
P(X > 1.23) \approx 0.1093
\]