A committee of 4 people is to be appointed from 3 officers of... | Agrigyan

• Probability of Non-occurrence of Independent Events:
• Given $n$ mutually independent events $A_1, A_2, \dots, A_n$ with probabilities:

$$P(A_i) = \frac{1}{i + 1}, \quad \text{for } 1 \le i \le n$$

• The complementary probability that event $A_i$ does not occur is:

$$P(A_i^c) = 1 - P(A_i) = 1 - \frac{1}{i + 1} = \frac{i}{i + 1}$$

• By independence of the events, the complements $A_1^c, A_2^c, \dots, A_n^c$ are also independent:

$$P\left(\bigcap_{i=1}^n A_i^c\right) = \prod_{i=1}^n P(A_i^c) = \prod_{i=1}^n \frac{i}{i + 1}$$

• Expanding the telescoping product:

$$P(\text{none occurs}) = \left(\frac{1}{2}\right) \times \left(\frac{2}{3}\right) \times \left(\frac{3}{4}\right) \times \dots \times \left(\frac{n-1}{n}\right) \times \left(\frac{n}{n + 1}\right)$$

• All intermediate numerator and denominator terms cancel out, leaving:

$$P(\text{none occurs}) = \frac{1}{n + 1}$$

• Conclusion: The probability that none of the $n$ events occurs is $\frac{1}{n+1}$ (Option 1).

Correct option: (1) Option 1

• Variance Stabilizing Transformations:
• When analyzing proportional data ($p$) where observations follow a binomial distribution, the variance is proportional to $p(1 - p)$, which depends heavily on the mean.
• Angular / Arcsine Transformation:
• The variance-stabilizing transformation for binomial proportions or percentages is the sine inverse transformation ($f(p) = \arcsin(\sqrt{p})$ or $\sin^{-1}\sqrt{p}$).
• This stabilizes the variance to a constant value of approximately $\frac{1}{4n}$ (in radians), satisfying the homoscedasticity assumption of ANOVA.
• Other Transformations:
• Square root transformation ($\sqrt{x}$ or $\sqrt{x + 0.5}$): Used for count data following a Poisson distribution.
• Logarithmic transformation ($\log x$): Used when standard deviation is proportional to the mean.
• Fisher's $z$-transformation: Used to normalize the sample correlation coefficient $r$.
• Conclusion: The appropriate transformation is Sine inverse transformation (Option 2).

Correct option: (2) Sine inverse transformation

• Neyman-Pearson Fundamental Lemma:
• In hypothesis testing theory, the Neyman-Pearson Lemma provides the necessary and sufficient condition for constructing the Most Powerful (MP) test of size $\alpha$ for testing a simple null hypothesis against a simple alternative hypothesis:

$$H_0: \theta = \theta_0 \quad \text{vs.} \quad H_1: \theta = \theta_1$$

• The optimal rejection region is defined by the likelihood ratio test statistic:

$$\Lambda(x) = \frac{L(x; \theta_1)}{L(x; \theta_0)} \gt k$$

• When the family of distributions possesses a Monotone Likelihood Ratio (MLR), this lemma generalizes to construct Uniformly Most Powerful (UMP) tests for one-sided composite hypotheses.
• Conclusion: The Neyman-Pearson lemma constructs the Most powerful test (Option 2).

Correct option: (2) Most powerful test