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

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
  1. 0.2
  2. 0.3
  3. 0.1
  4. 0.4

Solution

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

Asked in: MHT CET 2024 (16 May Shift 1)

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