To find [tex]\( P(X < -2.57) \)[/tex] for a standard normal random variable [tex]\( X \)[/tex], we need to determine the area under the standard normal curve to the left of [tex]\( -2.57 \)[/tex].
Here are the steps to solve the problem:
1. Standard Normal Distribution:
- The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. This distribution is denoted as [tex]\( N(0, 1) \)[/tex].
2. Z-Score:
- The value [tex]\( -2.57 \)[/tex] is a z-score. A z-score indicates how many standard deviations an element is from the mean. In this case, [tex]\( -2.57 \)[/tex] is 2.57 standard deviations below the mean.
3. Cumulative Distribution Function (CDF):
- To find the probability associated with a z-score, we use the cumulative distribution function (CDF) of the standard normal distribution. The CDF gives the probability that a standard normal random variable is less than or equal to a given value.
- Mathematically, [tex]\( P(X < z) = \Phi(z) \)[/tex], where [tex]\( \Phi(z) \)[/tex] is the CDF of the standard normal distribution.
4. Finding the Probability:
- We need to find [tex]\( \Phi(-2.57) \)[/tex]. This means we are looking for the area under the standard normal curve to the left of [tex]\( -2.57 \)[/tex].
The probability corresponding to the z-score [tex]\( -2.57 \)[/tex] is approximately [tex]\( 0.0050849257489910355 \)[/tex].
Therefore, [tex]\( P(X < -2.57) \approx 0.0050849257489910355 \)[/tex]. This tells us that the probability of [tex]\( X \)[/tex] being less than [tex]\( -2.57 \)[/tex] is about [tex]\( 0.51\)[/tex]% or approximately 0.005.
