x is a random variable that follows normal distribution with mean μ =

1.	x is a random variable that follows normal distribution with mean μ = 25 and standard deviation σ = 5. Find(i) P(x < 30)(ii) P(x > 18)(iii) P(25 < x < 30)
Answer» (i) P(x < 30) = P\((\frac{x-\mu}\sigma<\frac{30-25}5)\) = P\((\frac{x-\mu}\sigma<1)\) = 0.8413 (∵ \((\frac{x-\mu}\sigma∼N(0,1))\) (ii) P(x > 18) = 1 - P(x \(\leq18\)) = 1 - P\((\frac{x-\mu}\sigma\leq\frac{18-25}5)\) = 1 - P\((\frac{x-\mu}\sigma\leq1.4)\) = 1 - (P(x\(\leq\) 0) - P(-14 \(\leq\) x \(\leq\) 0)) = 1 - P(x \(\leq\) 0) + P(0 \(\leq\) x \(\leq\) 1.4) = 1 - P (x \(\leq\) 0) + (P(x \(\leq\)) - P(x \(\leq\)0)) = 1 - 2P(x \(\leq\) 0) + P(x \(\leq\) 1.4) = 1 - 2 x 0.5 + 0.9192 = 0.9192 (iii) P(25 < x < 30) = P(0 < \(\frac{x-\mu}{\sigma}<1\)) = P(x \(\leq\) 1) - P(x \(\leq\) 0) = 0.8413 - 0.5 = 0.3413

x is a random variable that follows normal distribution with mean μ = 25 and standard deviation σ = 5. Find(i) P(x < 30)(ii) P(x > 18)(iii) P(25 < x < 30)

Answer»

(i) P(x < 30) = P\((\frac{x-\mu}\sigma<\frac{30-25}5)\)

= P\((\frac{x-\mu}\sigma<1)\)

= 0.8413 (∵ \((\frac{x-\mu}\sigma∼N(0,1))\)

(ii) P(x > 18) = 1 - P(x \(\leq18\))

= 1 - P\((\frac{x-\mu}\sigma\leq\frac{18-25}5)\)

= 1 - P\((\frac{x-\mu}\sigma\leq1.4)\)

= 1 - (P(x\(\leq\) 0) - P(-14 \(\leq\) x \(\leq\) 0))

= 1 - P(x \(\leq\) 0) + P(0 \(\leq\) x \(\leq\) 1.4)

= 1 - P (x \(\leq\) 0) + (P(x \(\leq\)) - P(x \(\leq\)0))

= 1 - 2P(x \(\leq\) 0) + P(x \(\leq\) 1.4)

= 1 - 2 x 0.5 + 0.9192

= 0.9192

(iii) P(25 < x < 30) = P(0 < \(\frac{x-\mu}{\sigma}<1\))

= P(x \(\leq\) 1) - P(x \(\leq\) 0)

= 0.8413 - 0.5

= 0.3413

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