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If on an average $1$ ship out of $10$ do not arrive safely to ports. Find the mean and the standard deviation of ship returning safely out of a total of $500$ ships

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Toolbox:
  • A random variable $X$ is said to follow a binomial distribution of its probability mass function is given by
  • $P(X=x)=p(x)=\left\{\begin{array}{1 1}nC_xp^xq^{n-x},&x=0,1.......n\\0,&otherwise\end{array}\right.$
  • Constants of Binomial Distribution :
  • Mean = np
  • Variance = npq
  • Standard deviation =$\sqrt{ variance }=\sqrt{ npq}$
  • In a bionomial distribution Mean > Variance.
  • The parameters of the distribution are $n,p\quad X\sim B(n,p)$
Step 1:
Let $X$ be the random variable denoting the numbers of ships out of 500 that return safely to the port.
The probability of a ship not returning safely=$\large\frac{1}{10}$$=q$
$\therefore p=1-\large\frac{1}{10}=\frac{9}{10}$
$X\sim B(500,\large\frac{9}{10})$
Step 2:
The mean=np
$\qquad\quad\;\;=500\times \large\frac{9}{10}$
$\qquad\quad\;\;=450$
The standard deviation =$\sqrt{npq}$
$\qquad\qquad\qquad\qquad=\sqrt{500\times \large\frac{9}{10}\times \frac{1}{10}}$
$\qquad\qquad\qquad\qquad=\sqrt{45}$
$\qquad\qquad\qquad\qquad=3\sqrt{5}$
answered Sep 18, 2013 by sreemathi.v
 

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