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Identify binomial model and compute moments

UpstartData Scientist · HR Screen
EasyData ScientistHR ScreenStatistics & Math

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This question tests the ability to identify a binomial process, express its probability mass function, and derive its fundamental moments (mean and variance).

Scenario

Suppose you are analyzing the success rate of a new marketing campaign. You observe 10 independent users, and for each user, there is a fixed probability p = 0.2 that they will click on the advertisement. Let X be the random variable representing the total number of clicks observed from these 10 users.

Tasks

  1. Identify the name of the probability distribution that describes X and list the necessary assumptions for this model to hold.
  2. Write the formula for the Probability Mass Function (PMF) of X for a general case where there are n trials and probability p.
  3. Calculate the expected value E[X] and the variance Var[X] for the specific case where n = 10 and p = 0.2.

Example

If n = 3 and p = 0.5, the possible values for X are {0, 1, 2, 3}. The probability of X = 2 is calculated as C(3, 2) * (0.5)^2 * (0.5)^1 = 3 * 0.25 * 0.5 = 0.375.

Constraints

  • Assume each user's action is independent and the probability p remains constant across all users.
  • Provide the exact symbolic formulas before substituting the numerical values.

Hints

Hint 1

Recall that the distribution describes the number of successes in a fixed number of independent Bernoulli trials.

Hint 2

The PMF formula involves the binomial coefficient `C(n, k)`, the probability of success `p` raised to `k`, and the probability of failure `(1-p)` raised to `(n-k)`.

Hint 3

The expectation is `n * p` and the variance is `n * p * (1-p)`.

Follow-up questions

  • How would the distribution change if the probability of success `p` were not constant for each user?
  • If the number of trials `n` is very large and `p` is very small, which distribution can be used to approximate the binomial distribution?
  • What is the physical interpretation of the variance in this context—what does a high variance tell us about the predictability of the campaign results?