Determine Probability in a Normal Distribution

Solution & Explanation

To determine the probability of observing a value of 1000 in a standard normal distribution with a mean ($\mu$) of 0 and a standard deviation ($\sigma$) of 1, we need to understand the properties of the normal distribution.

Understanding the Normal Distribution:

• Mean ($\mu$): The average or central value of the distribution. Here, it is 0.
• Standard Deviation ($\sigma$): Measures the dispersion or spread of the distribution. Here, it is 1.
• Standard Normal Distribution: A normal distribution with $\mu = 0$ and $\sigma = 1$.

Probability of a Specific Value in a Continuous Distribution:

• In continuous distributions, the probability of observing a specific value is technically 0. This is because there are infinitely many possible values the variable can take.
• Instead, probabilities are calculated over intervals or ranges of values.

Calculating the Z-Score:

• The Z-score indicates how many standard deviations an element is from the mean.

• Formula for Z-score:
  $Z = \frac{X - \mu}{\sigma}$
  Where:

  • $X$ is the value of interest (1000 in this case).
  • $\mu$ is the mean (0).
  • $\sigma$ is the standard deviation (1).

• For $X = 1000$: $Z = \frac{1000 - 0}{1} = 1000$

Interpreting the Z-Score:

• A Z-score of 1000 means the value is 1000 standard deviations away from the mean.
• Given that 99.7% of data in a normal distribution lies within 3 standard deviations from the mean, a Z-score of 1000 is astronomically high.

Probability of Observing a Value of 1000:

• Practically Zero: The probability of observing a value of 1000 or higher is less than $10^{-200}$, which is essentially zero.
• Why?: The tails of the normal distribution drop off rapidly, and the probability mass is concentrated around the mean.
• In Practice: It's more useful to consider the probability of a value within a range rather than an exact value.

Conclusion:

• For a standard normal distribution, the probability of observing a value as extreme as 1000 is virtually zero.
• This scenario highlights the importance of understanding the distribution's properties and the concept of Z-scores in assessing probabilities.