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Question #41713

Difficulty: 8/10 | 🟢 Free

The Question

Let $p = \lim_{n \to \infty} \sum_{k=1}^n \frac{k^2}{n^3 + k^2 \sin(k)}$ . A biased coin with probability of heads $P(H) = \frac{1}{4}$ is tossed repeatedly. Let $N$ be the random variable representing the number of tails obtained before exactly 3 heads appear. Let $X$ be the number of successes in $N$ independent Bernoulli trials, where the probability of success in each trial is $p$ . Find the variance of the random variable $Z = X + \frac{N}{3}$ .

Options

A.
6
B.
10
C.
18
D.
24

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Conditional Probability Constraints|Bernoulli Trials Logic|Variance Invariance under Translation|Squeeze Theorem (Sandwich Theorem)

Why This Question Is Tricky

Common Mistakes

The question tests the Squeeze Theorem for limits and the Law of Total Variance in conditional probability.

The core trap is the 'Variance Invariance under Translation' fallacy.

Students often assume $Var(X + cN) = Var(X) + c^2 Var(N)$ , ignoring the covariance between the random number of trials $N$ and the binomial successes $X$ .

Alternatively, they treat $N$ as a constant expected value, yielding $Var(X+N/3) = Var(X) = 6$ .

By forcing the translation to be a function of the random variable $N$ itself, the problem demands rigorous application of $Var(Z) = E[Var(Z|N)] + Var(E[Z|N])$ , distinguishing rote memorizers from those with deep conceptual clarity.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Probability - Class 12
Concept	Conditional Probability Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free