Data Science Interview Prep: Q68

Probability of Exactly k Exceedances in N i.i.d. Normal Samples.(Category: Probability)

You draw N i.i.d numbers from a normal distribution with mean μ and standard deviation σ. What’s the probability that exactly k of these numbers are greater than some threshold Y?

Solution:

Let us understand what i.i.d stands for.

i.i.d. stands for independent and identically distributed:This refers to a set of random variables that share the same probability distribution and do not influence one another.

• Identically distributed: Each random variable in the set is drawn from the same probability distribution with identical parameters. For example, if the variables represent heights of people from a population, they all follow the same distribution (e.g., a normal distribution with the same mean, say 170 cm, and variance, say 25 cm²).

• Independent: The value of one random variable provides no information about the others. For instance, knowing the outcome of one coin flip doesn’t affect the next. Mathematically, for random variables X₁, X₂, X₃,…….,X_n, independence means:

  P(X_i, X_j) = P(X_i) × P(X_j) for any i ≠ j,

  meaning the joint probability of any pair is the product of their individual probabilities.

• Importance: The i.i.d. assumption is essential for many statistical and machine learning methods, such as hypothesis testing, regression, and various algorithms. It simplifies analysis and ensures reliable results by assuming data points are independent and drawn from the same distribution.

Now, let’s solve the problem.

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Reference:

• i.i.d draws from a Normal Distribution: https://math.stackexchange.com/questions/3788879/multiple-i-i-d-draws-from-normal-distribution