Paper 2, Section I, F

Probability | Part IA, 2021

Let XX be a continuous random variable taking values in [0,3][0, \sqrt{3}]. Let the probability density function of XX be

fX(x)=c1+x2, for x[0,3]f_{X}(x)=\frac{c}{1+x^{2}}, \quad \text { for } x \in[0, \sqrt{3}]

where cc is a constant.

Find the value of cc and calculate the mean, variance and median of XX.

[Recall that the median of XX is the number mm such that P(Xm)=12.]\left.\mathbb{P}(X \leqslant m)=\frac{1}{2} .\right]

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