GATE 2026 DA – Question 28
Suppose a random variable $Z$ follows $Normal(\mu = 0, \sigma^2 = 1)$ distribution with probability density function $g(z)$ and cumulative distribution function $G(z)$. Another random variable $Y$ follows $t_1$ distribution with probability density function $h(y)$ and cumulative distribution function $H(y)$. Let $c$ be the positive real number for which $g(c) = h(c)$.
Which of the following statements is/are correct?
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Correct answer: (A) $G(0) = H(0)$; (C) $G(-c) < H(-c)$
Explanation
The $t_1$ distribution is the Cauchy distribution, with $h(y) = \frac{1}{\pi(1 + y^2)}$, and it has heavier tails than the normal one. Both densities are symmetric about 0, so $G(0) = H(0) = 0.5$ (A is true). At 0 the normal density is higher, $g(0) = 0.399 > h(0) = 0.318$, so D is false. The densities cross at $c \approx 1.85$, beyond which the Cauchy density is higher. Then $G(c) \approx 0.968$ and $H(c) \approx 0.842$, so $G(c) > H(c)$ and B is false. By symmetry, $G(-c) = 1 - G(c) \approx 0.032$ and $H(-c) \approx 0.158$, so $G(-c) < H(-c)$ and C is true.