Pretty standard setup but the variance follow-up is where people slip up.
Define the random variable X as 1 for heads and 0 for tails, then compute E[X] = p and Var(X) = p(1-p) using the Bernoulli distribution properties. Explain each step clearly, showing the calculation and connecting it to the unfair coin scenario.
Pro tip: Mention that this is a Bernoulli trial and that the variance is maximized when p=0.5, which is relevant for A/B testing and product metrics at Snapchat.
Let X be the outcome of the coin flip, where X=1 for heads and X=0 for tails. This binary encoding simplifies the expected value and variance calculations.
Use the definition E[X] = sum(x * P(X=x)) = 1*p + 0*(1-p) = p. Explain that the expected value is simply the probability of heads.
Use Var(X) = E[X^2] - (E[X])^2. Since X^2 = X for binary variables, E[X^2] = p, so Var(X) = p - p^2 = p(1-p).
Discuss that the expected value is p and variance is p(1-p). Note that variance is highest at p=0.5 and decreases as p approaches 0 or 1.
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Geometric distribution, P(T=3) = (1-p)^2 * p.
First, identify the distribution as geometric by explaining that it models the number of independent Bernoulli trials until the first success. Then, state the probability mass function and compute P(X=3) = (1-p)^2 * p, assuming a fair coin p=0.5, which gives 0.125. Finally, connect this to product analytics by discussing how such distributions model user behavior like time-to-first-action.
Pro tip: Mention that the geometric distribution is memoryless, meaning past failures don't affect future probabilities—this is a key property that often comes up in analytics and can impress interviewers.
Recognize that the number of flips until the first heads is a geometric distribution with success probability p=0.5.
Write the PMF: P(X=k) = (1-p)^{k-1} * p for k=1,2,3,...
Plug in k=3 and p=0.5: P(X=3) = (0.5)^2 * 0.5 = 0.125.
Note that the geometric distribution is memoryless: given no success yet, the remaining number of flips still follows the same geometric distribution.
Discuss how this models time-to-first-event (e.g., first snap sent) and why understanding such distributions helps in analyzing user engagement.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Start by clearly defining the geometric distribution and its parameter p, then derive the expected value using the formula E[X] = 1/p. If needed, show the derivation using the memoryless property or summation of probabilities to demonstrate understanding.
Pro tip: Connect the concept to real-world applications at Snapchat, such as modeling user engagement or ad clicks, to show business impact. Also, be prepared to discuss variance and the memoryless property, as interviewers often probe deeper.
Let X be the number of flips until the first heads. Assume each flip is independent with probability p of heads.
Recognize that X follows a geometric distribution with parameter p.
For a geometric distribution, E[X] = 1/p.
Show derivation using the memoryless property or by summing the infinite series: E[X] = sum_{k=1}^∞ k * (1-p)^{k-1} * p = 1/p.
Mention variance (1-p)/p^2, the memoryless property, and potential applications in product analytics.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the null and alternative hypotheses: H0: p=0.5 (fair coin) vs H1: p<0.5 (heads rarer). Then, define the test statistic as the number of flips until heads (T), and compute the p-value as the probability of observing T ≤ 3 under H0, using the geometric distribution. Finally, interpret the p-value in context, noting that it does not prove the coin is fair or unfair, and discuss the limitations of a single observation.
Pro tip: Emphasize that with only one observation, the p-value is a crude measure; a data scientist should advocate for collecting more data to make reliable inferences. Also, mention that the geometric distribution assumes independence and constant probability, which are key assumptions to check.
Define the null hypothesis H0: p=0.5 (coin is fair) and the alternative H1: p<0.5 (heads are rarer than expected).
The test statistic is T, the number of flips until the first heads. Under H0, T follows a geometric distribution with success probability p=0.5.
Since the alternative is one-sided (p<0.5), the p-value is P(T ≤ 3 | p=0.5) = 1 - P(T > 3) = 1 - (0.5)^3 = 0.875.
A p-value of 0.875 is large, so we fail to reject H0. However, with a single observation, the test has low power; recommend collecting more data for a reliable conclusion.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.