If the p.m.f. of a r.v. $\mathrm{X}$ is given by $\mathrm{P}(\mathrm{X}=x)=\frac{\left(\begin{array}{l}5 \ x\end{array} ight)}{2^{5}}$ if $x=0,1,2, \ldots \ldots 5$ $=0$ otherwise then which of the following is not true?

Question

If the p.m.f. of a r.v. $\mathrm{X}$ is given by $\mathrm{P}(\mathrm{X}=x)=\frac{\left(\begin{array}{l}5 \ x\end{array}ight)}{2^{5}}$ if $x=0,1,2, \ldots \ldots 5$ $=0$ otherwise then which of the following is not true?

Aicharya · Accepted Answer

$\begin{aligned} P(X \leq 2) &=P(X=0)+P(X=1)+P(X=2) \ &=\frac{{ }^{5} C_{0}}{2^{5}}+\frac{{ }^{5} C_{1}}{2^{5}}+\frac{{ }^{5} C_{2}}{2^{5}}=\frac{1+5+10}{2^{5}}=\frac{16}{2^{5}} \ P(X \geq 4) &=P(X=4)+P(X=5) \ &=\frac{C_{4}}{2^{4}}+\frac{C_{6}}{2^{6}}=\frac{5+1}{2^{5}}=\frac{6}{2^{5}} \end{aligned}$ $\begin{aligned} \mathrm{P}(\mathrm{X} \leq 3) &=\mathrm{P}(\mathrm{X}=0)+\mathrm{P}(\mathrm{X}=1)+\mathrm{P}(\mathrm{X}=2)+\mathrm{P}(\mathrm{X}=3) \ &=\frac{{ }^{5} \mathrm{C}_{0}}{2^{5}}+\frac{{ }^{5} \mathrm{C}_{1}}{2^{5}}+\frac{{ }^{5} \mathrm{C}_{2}}{2^{5}}+\frac{{ }^{5} \mathrm{C}_{3}}{2^{5}}=\frac{1+5+10+10}{2^{5}}=\frac{26}{2^{5}} \ \mathrm{P}(\mathrm{X} \geq 3) &=\mathrm{P}(\mathrm{X}=3)+\mathrm{P}(\mathrm{X}=4)+\mathrm{P}(\mathrm{X}=5) \ &=\frac{{ }^{5} \mathrm{C}_{3}}{2^{5}}+\frac{{ }^{5} \mathrm{C}_{4}}{2^{5}}+\frac{{ }^{5} \mathrm{C}_{5}}{2^{5}}=\frac{10+5+1}{2^{5}}=\frac{16}{2^{5}} \ \mathrm{P}(\mathrm{X} \leq 1) &=\mathrm{P}(\mathrm{X}=0)+\mathrm{P}(\mathrm{X}=1) \ &=\frac{{ }^{5} \mathrm{C}_{0}}{2^{5}}+\frac{{ }^{5} \mathrm{C}_{1}}{2^{5}}=\frac{1+5}{2^{5}}=\frac{6}{2^{5}} \end{aligned}$

If the p.m.f. of a r.v. $\mathrm{X}$ is given by $\mathrm{P}(\mathrm{X}=x)=\frac{\left(\begin{array}{l}5 \\…

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