A random variable \(X\) takes the values \(0,1,2,3\) and its mean is 1.3. If \(\mathrm{P}(\mathrm{X}=3)=2 \mathrm{P}(\mathrm{X}=1)\) and \(P(X=2)=0.3\), then \(P(X=0)\) is

Given,
\(\mathrm{P}(\mathrm{X}=3)=2 \mathrm{P}(\mathrm{X}=1) \text { and } \mathrm{P}(\mathrm{X}=2)=0.3...(i)\)
Now, mean \(=1.3\)
\(\begin{aligned}
& \therefore \quad 0 \times \mathrm{P}(\mathrm{X}=0)+1 \times \mathrm{P}(\mathrm{X}=1)+2 \times \mathrm{P}(\mathrm{X}=2) \\
& +3 \times \mathrm{P}(\mathrm{X}=3)=1.3 \\
& \Rightarrow 7 \mathrm{P}(\mathrm{X}=1)=0.7 \\
& \text {...[From (i)] } \\
& \Rightarrow \mathrm{P}(\mathrm{X}=1)=0.1 \\
& \text { Also, } \mathrm{P}(\mathrm{X}=0)+\mathrm{P}(\mathrm{X}=1)+\mathrm{P}(\mathrm{X}=2) \\
& +\mathrm{P}(\mathrm{X}=3)=1 \\
& \Rightarrow \mathrm{P}(\mathrm{X}=0)+3 \mathrm{P}(\mathrm{X}=1)=0.7 \quad \ldots[\text { From (i) }] \\
& \Rightarrow \mathrm{P}(\mathrm{X}=0)+0.3=0.7 \\
& \Rightarrow \mathrm{P}(\mathrm{X}=0)=0.4
\end{aligned}\)