Q.1
Mode of F distribution with V1 (2) and V2 degrees of freedom is given as
- 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:
- 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)]
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