Probability Distributions

Here $\lambda = \cfrac{1}{500}\times 10 = 0.02$
Thus probability that  lot is not defective is $=P(X=0)=\cfrac{e^{-0.002}(.0020^0}{0!}=e^{-.002}=0.9802$
Hence number of no defective lots out of $10,000$ is $=.9802\times 10,000=9802$

Here $\lambda = 0.001$
Hence number of day out of 1000 days without accident is $1000\times P(X=0)=1000\times e^{-0.001}$

we are given $n=100$
let $p=$ probability of a defective bulb $=5$%$=0.05$
$\therefore m=$ mean number of defective bulbs in a box of $100=np=100\times 0.05=5$
Since p is small , we can use poison's distribution.
Probability of $x$ defective bulbs in a box of $100$ is
$\displaystyle P\left( X=x \right) =\frac { { e }^{ -m }{ m }^{ x } }{ x! } =\frac { { e }^{ -5 }{ 5 }^{ x } }{ x! } ,x=0,1,2...$
Probability that is box will fail to meet the guarented quality is $\displaystyle P\left( X>10 \right) =1-P\left( X\le 10 \right) =1-\sum _{ x=0 }^{ 10 }{ \frac { { e }^{ -5 }{ 5 }^{ x } }{ x! }  } =1-{ e }^{ -5 }\sum _{ x=0 }^{ 10 }{ \frac { { 5 }^{ x } }{ x! }  } $

Here $\lambda = 3$
Hence Number of drivers with no accident out of 1000 is $=1000\times P(X=0)=1000\times e^{-3}=1000\times 0.0498=49.8$

By using Poisson distribution, 
$ P(x;\mu)=\dfrac { { e }^{ -\mu }{ \mu }^{ x } }{ x! } $
$ \mu $: The mean number of successes that occur in a specified region(parameter)
x: The actual number of successes that occur in a specified region.
P(x; $ \mu $): The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is $ \mu $
Here,$ \mu $  = 2 
x = 2 (exact 2 workers)
$ P(2;2)=\dfrac { { e }^{ -2 }{ 2 }^{2 } }{ 2! } $

Here $\lambda = \cfrac{0.1}{100}\times 500=0.5 $
Hence number of boxes out of 100 which contain at least one defective bottle is,
$=100\left(1-P(X=0)\right)=100\left(1-e^{-0.5}\right)$ 

Here P.D parameter $\lambda = \cfrac{2}{100}\times 100=2$
Hence probability that 3 or more people are left handed is $=1-P(X=0)-P(X=1)-P(X=2)$
$=1-e^{-2}-\cfrac{e^{-2}2}{1!}-\cfrac{e^{-2}2^2}{2!}=1-5e^{-2}$

P.D parameter $\lambda = \cfrac{2}{50000}\times 100000 = 4$
Thus probability that in a year there is no suicide is $=P(X=0)=e^{-4}$

Here P.D parameter $\lambda = \cfrac{5}{2000}\times 10000=25$
Hence probability that exactly 10 houses will get damaged $=P(X = 10) = \cfrac{e^{-25}(25)^{10}}{10!}$

$\displaystyle p=\frac { 0.1 }{ 100 } =0.001,n=500\ \lambda =np=500\times 0.001=0.5\ N=100$
We know that $\displaystyle p\left( r \right) =e^{ -1 }\frac { { \lambda  }^{ r } }{ r! } $
Number of boxes containing no defective bottle 
$\displaystyle =N.P.\left( r=0 \right) =1000\times { e }^{ 0.5 }\frac { \left( 0.5 \right) ^{ 0 } }{ 0! } =1000\times { e }^{ -0.5 }$

Here P.D parameter $\lambda = \cfrac{2}{100}\times 100=2$
Hence number of probability that 3 blade are defective is $=P(X=3)=\cfrac{e^{-2}(2)^3}{3!}=\cfrac{4}{3}e^{-2}=\cfrac{4}{3}\times .1353=.1804$

By using Poisson distribution, 
$ P(x;\mu)=\dfrac { { e }^{ -\mu }{ \mu }^{ x } }{ x! } $
$ \mu $: The mean number of successes that occur in a specified region(parameter)
x: The actual number of successes that occur in a specified region.
P(x; $ \mu $): The Poisson probability that exactly x successes occur in a Poisson experiment, when the mean number of successes is $ \mu $
the average a submarine on patrol sights 6 enemy ships per hour, so for 2 hours
Here, $ \mu =  6 \times 2 = 12 $
x = 4 ( ships)
$ P(4;12)=\dfrac { { e }^{ -12 }{12 }^{4 } }{ 4! }$

Random Arrival process is a Poisson process
Given by
$P(k) = \dfrac{e^{-\lambda} \lambda^x}{x!}$
$given, \lambda = 1/5 min^{-1}= 12 s^{-1}$
Room is not full, if No.of patients $< 12$
Hence 
$P$(room is full) $= 1 - (P(1)+ . . . . +P(11))$
$=1-\displaystyle \sum _{{x}=0}^{11}\frac{{e}^{-12}(12)^{{x}}}{ x!}$

The probability of sighting at least two ships
=1-(Probability of sighting atmost 1 ships)
$=1-e^{-\lambda}-\dfrac{\lambda^{1}e^{-\lambda}}{1!}$
Here $\lambda=2$
Hence 
$P=1-e^{-\lambda}-\dfrac{\lambda^{1}e^{-\lambda}}{1!}$
$=1-e^{-2}-2e^{-2}$
$=1-3e^{-2}$.