Let a continuous random variable X have pdf \(\rm f(x)=\left\{\begin{matrix}\frac{3x^2}{θ ^2},& 0\le x \le θ\\\ 0,& \rm otherwise\end{matrix}\right.\) for some θ > 0, the mode of X is :

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Question

This question was previously asked in

SSC CGL Tier-II (JSO) 2022 Official Paper (Held On: 4 March 2023)

Answer 1

The correct answer is θ.

Key Points

The mode of a probability distribution is the value of the random variable where the probability density function (pdf) reaches its maximum. The pdf for the random variable X given here is a continuous function on the interval [0, θ], and is 0 everywhere else.
This particular pdf is a monotonically increasing function of x on the interval [0, θ] (since x^2 is an increasing function on [0, θ] and θ^2 is a positive constant). This means that the function f(x) increases as x increases within this interval.

Hence, the mode of X, the point where the pdf f(x) is at its maximum, is at x = θ.