\(\mathrm{P}(\mathrm{Z}>\mathrm{z}) =0.5832 \)
\(\mathrm{P}(-\mathrm{z}<\mathrm{Z}<0) =0.5832-0.5=0.0832 \)
\(\mathrm{z} =-0.21 \)

\(\mathrm{P}(\mathrm{Z}>\mathrm{z}) =0.8508 \)
\(\mathrm{P}(-\mathrm{z}<\mathrm{Z}<0) =0.8508-0.5=0.3508 \)
\(\mathrm{z} =-1.04 \)

\(\mathrm{P}(\mathrm{Z}<\mathrm{z}) =0.0793 \)
\(0.5-\mathrm{P}(0<\mathrm{Z}<\mathrm{z}) =0.0793 \)
\(\mathrm{P}(0<\mathrm{Z}<\mathrm{z}) =0.5-0.0793=0.4207 \)
\(\mathrm{z} =-1.41\)

\(P(Z>z) =0.8413 \)
0.5+P(-z<Z<0)  =0.8413
P(-z<Z<0)  =0.3413
z  =-1

\(\mu=1500, \sigma=200\)
\(\mathrm{Z}=\frac{X-\mu}{\sigma}\)
\(=\frac{1000-1500}{200}=-2.5 \)

\(P(X<1000) =P(Z<-2.5) \)
=0.5-P(0<Z<2.5) 
 =0.5-0.4938=0.0062

\(\mathrm{n} =4, \mathrm{p}=\frac{21}{100}=0.21 \)
\(\mathrm{P}(\mathrm{X} =4)=4 \mathrm{C}_4(0.21)^4=0.001944 \)

\(\mu=172, \sigma^2=25 \Rightarrow \sigma=5\)
\(Z=\frac{X-\mu}{\sigma}=\frac{165-172}{5}=-1.4\)
\(Z=\frac{181-172}{5}=1.8\)

\(\mathrm{P}(165<\mathrm{X}<181)=\mathrm{P}(-1.4<\mathrm{Z}<1.8) \)
\(=\mathrm{P}(-1.4<\mathrm{Z}<0)+\mathrm{P}(0<\mathrm{Z}<1.8) \)
=0.4192+0.4641=0.8833=0.883

\( \mu=180000, \sigma=10000 \)
\( Z=\frac{X-\mu}{\sigma}=\frac{165000-180000}{10000}=-1.5 \)

\(Z=\frac{175000-180000}{1000}=-0.5\)
\( \mathrm{P}(165000<\mathrm{X}<175000) \)
\(=\mathrm{P}(-1.5<\mathrm{Z}<-0.5) \)
\(=\mathrm{P}(-1.5<\mathrm{Z}<0)-\mathrm{P}(-0.5<\mathrm{Z}<0) \)
=0.4332-0.1915=0.2417=0.242

μ = 70, σ = 10
\(\mathrm{Z}=\frac{x-\mu}{\sigma}=\frac{72-70}{10}=0.2\)
\(Z=\frac{84-70}{10}=1.4\)

\(\mathrm{P}(72<\mathrm{X}<\mathrm{8} 4) =\mathrm{P}(0.2<\mathrm{Z}<1.4) \)
\( =\mathrm{P}(0<\mathrm{Z}<1.4)-\mathrm{P}(0<\mathrm{Z}<0.2) \)
 =0.4192-0.0793=0.339  =0.34

சராசரி = \(\lambda=n p=15 \times \frac{40}{100}=6\)

சராசரி = np = 4, மாறுபாடு = npq = 4/3
\(\Rightarrow 4 q =\frac{4}{3} \Rightarrow q=\frac{1}{3} \)
\(p=1-q =1-\frac{1}{3}=\frac{2}{3} \)
\(n p =4 \Rightarrow n\left(\frac{2}{3}\right)=4 \Rightarrow n=6 \)
\(P(X \geq 5) =P(X=5)+P(X=6) \)
\( =6 C_5\left(\frac{2}{3}\right)^5\left(\frac{1}{3}\right)+6 C_6\left(\frac{2}{3}\right)^6 \)
\( =\frac{2^5}{3^6}[6+2]=4\left(\frac{2}{3}\right)^6 \)

\( p=2 q \Rightarrow p=2(1-p)=2-2 p \)
\( 3 p=2 \Rightarrow p=\frac{2}{3} \)
\( q=1-p=1-\frac{2}{3}=\frac{1}{3} \)
\( \mathrm{n}=6 \)
\( P(X \geq 4)=\mathrm{P}(\mathrm{X}=4)+\mathrm{P}(\mathrm{X}=5)+\mathrm{P}(\mathrm{X}=6) \)
\( =6 C_4\left(\frac{2}{3}\right)^4\left(\frac{1}{3}\right)^2+6 C_5\left(\frac{2}{3}\right)^5\left(\frac{1}{3}\right)+6 C_6\left(\frac{2}{3}\right)^6 \)
\( =\frac{2^4}{3^6}\left[6 C_2+6 C_1 2+2^2\right] \)
\(=\frac{16}{729}[15+12+4]=\frac{496}{729}\)

\( \mathrm{p}=\frac{2}{100}, \lambda=\mathrm{np}=50\left(\frac{2}{100}\right)=1 \)
\(P(X \leq 2) =\mathrm{P}(\mathrm{X}=0)+\mathrm{P}(\mathrm{X}=1)+\mathrm{P}(\mathrm{X}=2) \)
\(=\mathrm{e}^{-1}+\mathrm{e}^{-1}(1)+\frac{e^{-1}(1)}{2 !} \)
\( =e^{-1}\left[2+\frac{1}{2}\right]=2.5 \mathrm{e}^{-1}\)

\( \mu=10 \)
\( 2 \sigma^2=72 \Rightarrow \sigma^2=36 \)
\( N\left(\mu, \sigma^2\right)=(10,36) \)

\(f(x)=\frac{1}{\sigma \sqrt{2 \pi}} \exp \left\{\frac{-1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right\}\)
வளைவு மாற்றப்புள்ளி \(x=\mu \pm \sigma\)
\(\mathrm{f}(x)  =\frac{1}{\sigma \sqrt{2 \pi}} \exp \left\{\frac{-1}{2}\left(\frac{\mu \pm \sigma-\mu}{\sigma}\right)^2\right\} \)
\(=\frac{1}{\sigma \sqrt{2 \pi}} e^{-\frac{1}{2}}\)

\(X \sim N(9,81)\)
இங்கு \(\mu=9\) மற்றும் \(\sigma^2=81 \Rightarrow \sigma=9\)
\(Z=\frac{X-\mu}{\sigma}=\frac{X-9}{9}\)