Which of the following statements are TRUE?

(A) The variable t of t-distribution ranges from $-\infty$ to $\infty$.

(B) The probability curve of the t-distribution is symmetric about the line $t=0$

(C) The variance of the t-distribution is greater than one.

(D) As the number of degrees of freedom decreases, the t-distribution curve moves closer to the standard normal probability curve.

Choose the correct answer from the options given below:

The variable t of t-distribution extends from $-\infty$ to $\infty$, similar to the normal distribution. The distribution has long tails on both sides that extend indefinitely.

Statement (A) is TRUE.

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The t-distribution curve is symmetric about the line $t=0$. Drawing a vertical line at $t=0$ shows the left half is a mirror image of the right half. The distribution has a mean of 0 and is balanced equally on both sides.

Statement (B) is TRUE.

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The variance of a t-distribution with $v$ degrees of freedom is:

$\text{Variance} = \frac{v}{v-2}$ for $v > 2$

For $v = 3$:

$\text{Variance} = \frac{3}{3-2}$

$= \frac{3}{1}$

$= 3$

For $v = 10$:

$\text{Variance} = \frac{10}{10-2}$

$= \frac{10}{8}$

$= 1.25$

For $v = 30$:

$\text{Variance} = \frac{30}{28}$

$= 1.07$

The variance is always greater than 1 for $v > 2$. As degrees of freedom increase, the variance approaches 1, which is the variance of a standard normal distribution.

Statement (C) is TRUE.

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As the number of degrees of freedom decreases, the t-distribution curve moves away from the standard normal curve, developing fatter tails. The correct relationship is:

More degrees of freedom → curve approximates standard normal distribution

Fewer degrees of freedom → curve has heavier tails and differs more from standard normal

Statement (D) is FALSE.

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The true statements are (A), (B), and (C).