# The probability distribution of a random variable X, where k is a constant is given below :,Determine (a) the value of k(b) (c) Mean of the variable X.

Given:

$\begin{array}{|c|c|c|} \hline \mathrm{x}_{i} & \mathrm{P}_{{i}} & \mathrm{P}_{{i}} \mathrm{x}_{i} \\ \hline 0 & 0.1 & 0\\ \hline 1 & \mathrm{k} & k \\\hline 2 & 2 \mathrm{k} & 4\mathrm{k}\\ \hline 3 & 3 \mathrm{k}&9 \mathrm{k} \\ \hline \end{array}$

i)

$\sum P_{i}=1 \\\\ 0.1+k+2k+3k=1 \\\\ 6k = 0.9 \\\\ k=\frac{3}{20}$

ii)

\begin{aligned} P(x \leq 2) &=P(0)+P(1)+P(2) \\ & = 0.1+k + 2k \\ &=\frac{1}{10}+\frac{3}{20}+\frac{6}{20} \\ & =\frac{11}{20} \end{aligned}

iii)

$\\ \text { Mean }= \sum P_{i} x_{i} \\ \\ =k+4k+9k \\ \\ =14 k\\\\=\frac{21}{10}$

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