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

$ k = \frac{2}{n(n+1)}$; $\mathrm{P}(X \leq n) = \frac{x(x+1)}{n(n+1)}, x =1,2,\ldots,n$.