The probability distribution of a discrete r.v. X is as follows :x123456P(X = x)k2k3k4k5k6k(a) Determine the value of k.(b) Find \(P(X\lt 4),\ P(2\lt X\lt 4),\ P(X\gt 3)\)

Q.5 (B) (iii) The probability distribution of a discrete r.v. X is as follows :

x	1	2	3	4	5	6
P(X = x)	k	2k	3k	4k	5k	6k

(a) Determine the value of k.
(b) Find \(P(X\lt 4),\ P(2\lt X\lt 4),\ P(X\gt 3)\)

Doubtly Asked 28 Oct 2026 Board exam · March 2026 hsc 12th Save WhatsApp Report

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Answer: \(k=\frac{1}{21}\); \(P(X\lt 4)=\frac{2}{7},\ P(2\lt X\lt 4)=\frac{1}{7},\ P(X\gt 3)=\frac{5}{7}\)

(a) \(\Sigma P(X=x)=1\Rightarrow k+2k+3k+4k+5k+6k=21k=1\Rightarrow k=\frac{1}{21}\).

(b) \(P(X\lt 4)=P(1)+P(2)+P(3)=6k=\frac{6}{21}=\frac{2}{7}\)

\(P(2\lt X\lt 4)=P(X=3)=3k=\frac{3}{21}=\frac{1}{7}\)

\(P(X\gt 3)=P(4)+P(5)+P(6)=15k=\frac{15}{21}=\frac{5}{7}\)

Doubtly AI · 28 Oct 2026 AI-verified · Oct 2026 Report

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