Estimating Days to Exceed a Threshold in Normal Distribution

Solution & Explanation

Understanding the Problem: The problem is asking us to determine the expected number of days it will take to draw a sample from a standard normal distribution (mean = 0, standard deviation = 1) that exceeds a value of 2.

Step-by-Step Breakdown:

1. Identify the Distribution:

   • We are dealing with a standard normal distribution, denoted as $X \sim N(0,1)$.
   • This implies that the mean ($\mu$) is 0, and the standard deviation ($\sigma$) is 1.

2. Calculate the Probability of Exceeding 2:

   • We need to find $P(X > 2)$.
   • Using the cumulative distribution function (CDF) of a standard normal distribution, we first find $P(X \leq 2)$.
   • $P(X \leq 2) = \Phi(2)$, where $\Phi$ is the CDF for the standard normal distribution.
   • From standard normal distribution tables or calculators, $\Phi(2) \approx 0.97725$.
   • Therefore, $P(X > 2) = 1 - \Phi(2) = 1 - 0.97725 = 0.02275$.

3. Determine the Expected Number of Days:

   • The process of drawing samples every day can be modeled as a series of Bernoulli trials, where each day is a trial and a success is obtaining a sample greater than 2.
   • The probability of success on any given day is $p = 0.02275$.
   • The expected number of trials (days) to get the first success in a series of independent Bernoulli trials is given by the reciprocal of the probability of success.
   • Therefore, the expected number of days, $E(D)$, is $\frac{1}{p} = \frac{1}{0.02275} \approx 43.956$.
   • Since we can't have a fraction of a day, we round up to the nearest whole number, resulting in an expected time of 44 days.

Conclusion: The expected number of days it will take to draw a sample from a standard normal distribution that exceeds a value of 2 is approximately 44 days. This calculation assumes each day is an independent trial with a constant probability of success.