A random variable \(X\) has the following probability distribution:\(x\)1234567\(P[X=x]\)\(k\)\(2k\)\(2k\)\(3k\)\(k^2\)\(2k^2\)\(7k^2+k\)Determine (a) \(k\) (b) \(P(X\lt 3)\) (c) \(P(X\gt 6)\) (d) \(P(0\lt X\lt 1)\)

Q.5 (B) (iii) A random variable \(X\) has the following probability distribution:

\(x\)	1	2	3	4	5	6	7
\(P[X=x]\)	\(k\)	\(2k\)	\(2k\)	\(3k\)	\(k^2\)	\(2k^2\)	\(7k^2+k\)

Determine (a) \(k\) (b) \(P(X\lt 3)\) (c) \(P(X\gt 6)\) (d) \(P(0\lt X\lt 1)\)

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

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Answer: (a) \(k=\frac{1}{10}\) (b) \(\frac{3}{10}\) (c) \(\frac{17}{100}\) (d) 0

(a) \(\sum P(x)=1\): \(k+2k+2k+3k+k^2+2k^2+7k^2+k=1\Rightarrow 10k^2+9k-1=0\Rightarrow(10k-1)(k+1)=0\). Since \(k\ge 0\), \(k=\frac{1}{10}\).

(b) \(P(X\lt 3)=P(1)+P(2)=3k=\frac{3}{10}\).

(d) \(X\) takes no value strictly between 0 and 1, so \(P(0\lt X\lt 1)=0\).

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

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