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  • The probability distribution of a random variable X, where k is a constant is given below : <img alt="P(X=x)=\left\{\begin{matrix} 0.1, &if &x=0 \\ kx^{2} &if &x=1 \\ kx&if &x=2 or x=3 \\ 0. &otherwise & \end{m

The probability distribution of a random variable X, where k is a constant is given below :

P(X=x)=\left\{\begin{matrix} 0.1, &if &x=0 \\ kx^{2} &if &x=1 \\ kx&if &x=2 or x=3 \\ 0. &otherwise & \end{matrix}\right.,

Determine (a) the value of k

(b) P(x \leq 2)

(c) Mean of the variable X.

 

 
 
 
 
 

Answers (1)

Given:

P(X=x)=\left\{\begin{matrix} 0.1, &if &x=0 \\ kx^{2} &if &x=1 \\ kx&if &x=2 or x=3 \\ 0. &otherwise & \end{matrix}\right.

\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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Safeer PP

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