The probability distribution of a discrete random variable $X$ is given by \(\mathrm{P}(X = x) = k

Question

The probability distribution of a discrete random variable $X$ is given by $\mathrm{P}(X = x) = kx$ where $x = 1, 2, 3, \dots n$. Find $k$ in terms of $n$.[2] Find, in terms of $x$ and $n$, an expression for $\mathrm{P}(X \leq x)$, the cumulative distribution function of $X$.[3]

Accepted Answer

**(a)** $k=\frac{2}{n(n+1)}$. **(b)** $\mathrm{F}(x)=\begin{cases}0,&x<1,\ \frac{\lfloor x\rfloor(\lfloor x\rfloor+1)}{n(n+1)},&1\leq x<n,\ 1,&x\geq n.\end{cases}$

Cumulative distribution function of a DRV

Cumulative distribution function of a DRV

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