Probability of accepting Null hypothesis Ho when itis false is... | Agrigyan

• Conditional Distribution of Independent Poisson Variates:
• Let $X \sim \text{Poisson}(m)$ and $Y \sim \text{Poisson}(n)$ be independent random variables.
• By the reproductive (additive) property of the Poisson distribution, the sum $Z = X + Y$ follows a Poisson distribution with parameter $(m + n)$:

$$Z = X + Y \sim \text{Poisson}(m + n)$$

• The conditional distribution of $X$ given $X + Y = k$ is given by Bayes' theorem:

$$P(X = x \mid X + Y = k) = \frac{P(X = x, Y = k - x)}{P(X + Y = k)} = \frac{\frac{e^{-m} m^x}{x!} \times \frac{e^{-n} n^{k - x}}{(k - x)!}}{\frac{e^{-(m + n)} (m + n)^k}{k!}}$$ $$P(X = x \mid X + Y = k) = \binom{k}{x} \left(\frac{m}{m + n}\right)^x \left(\frac{n}{m + n}\right)^{k - x}$$

• This is exactly the probability mass function of a Binomial distribution $\text{Bin}\left(k, \frac{m}{m + n}\right)$.
• Conclusion: The conditional distribution is Binomial (Option 3).

Correct option: (3) Binomial

• Rao-Blackwell Theorem in Point Estimation:
• The Rao-Blackwell theorem states that if $T$ is a sufficient statistic for a parameter $\theta$, and $U$ is any unbiased estimator of $\theta$, then the conditional expectation:

$$\phi(T) = E(U \mid T)$$

is an unbiased estimator of $\theta$ whose variance is strictly less than or equal to the variance of $U$ (i.e., $\text{Var}(\phi(T)) \le \text{Var}(U)$).

• Furthermore, by the Lehmann-Scheffé theorem, if the sufficient statistic $T$ is also complete, then $\phi(T)$ is the unique Uniformly Minimum Variance Unbiased Estimator (UMVUE).
• Other Options:
• Central Limit Theorem: Describes asymptotic normality of sample means.
• Cramér-Rao inequality: Provides a theoretical lower bound for the variance of unbiased estimators.
• Neyman-Pearson lemma: Constructs Most Powerful tests in hypothesis testing.
• Conclusion: Rao-Blackwell theorem (Option 3).

Correct option: (3) Rao-Blackwell theorem

• Mode of Snedecor's F-Distribution:
• For an $F$-distribution with $\nu_1$ numerator degrees of freedom and $\nu_2$ denominator degrees of freedom, the probability density function is unimodal for $\nu_1 \gt 2$.
• Differentiating the PDF with respect to $F$ and setting $\frac{df(F)}{dF} = 0$, the mode is found at:

$$\text{Mode}(F) = \frac{\nu_2(\nu_1 - 2)}{\nu_1(\nu_2 + 2)}, \quad \text{for } \nu_1 \gt 2$$

• Key Moments of F-Distribution:
• Mean: $E(F) = \frac{\nu_2}{\nu_2 - 2}$ (for $\nu_2 \gt 2$)
• Variance: $\text{Var}(F) = \frac{2\nu_2^2(\nu_1 + \nu_2 - 2)}{\nu_1(\nu_2 - 2)^2(\nu_2 - 4)}$ (for $\nu_2 \gt 4$)
• Conclusion: The mode is given by $\frac{\nu_2(\nu_1 - 2)}{\nu_1(\nu_2 + 2)}$ (Option 1).

Correct option: (1) 1V2(V1-2)NV4(V2+2)]