Question

10 0.05 20 0.2 25 0.1 30 0.2 0.16 0.05 0.1 a) P(x>6) b) P(x211) C) P(x < 10) d) P(x = 11 or x = 10) e) The mean for variable x f) The standard deviation for variable x. 115

Answer 1

Solution

Condition of probability distribution:

The sum of probability must be equal to 1

$\small \sum P(x)=1$

P(2)0.160.05+0.050.1 +0.2 +0.10.2

P(2) 1-0.86

P(2) 0.14

$\small \mathbf{x}$	2	6	10	11	16	20	25	30	$\small \mathbf{Total}$
$\small \mathbf{p\left (x \right )}$	0.14	0.16	0.05	0.05	0.1	0.2	0.1	0.2
$\small \mathbf{x*p\left (x \right )}$	0.28	0.96	0.50	0.55	1.60	4.00	2.50	6.00	$\small \mathbf{16.39}$
$\small \mathbf{x^{2}*p\left (x \right )}$	0.56	5.76	5.00	6.05	25.60	80.00	62.50	180.00	$\small \mathbf{365.47 }$

$\small {\color{Blue} \mathbf{a)}}\;P(x>6)=P(10)+P(11)+P(16)+P(20)+P(25)+P(30)$

$\small P(x>6)=0.05+ 0.05 +0.1 +0.2 +0.1+ 0.2$

$\small P(x>6)=\mathbf{{\color{Red} 0.70}}$

--------------------------------------------------------------------------------------------------------------

$\small {\color{Blue} \mathbf{b)}}\;P(x\geq 11)=P(11)+P(16)+P(20)+P(25)+P(30)$

$\small P(x\geq 11)=0.05 +0.1 +0.2 +0.1+ 0.2$

$\small P(x\geq 11)=\mathbf{{\color{Red} 0.65 }}$

--------------------------------------------------------------------------------------------------------------

$\small {\color{Blue} \mathbf{c)}}\;P(x<10)=P(2)+P(6)$

$\small P(x<10)=0.14 +0.16$

$\small P(x<10)=\mathbf{{\color{Red} 0.30}}$

--------------------------------------------------------------------------------------------------------------

$\small {\color{Blue} \mathbf{d)}}\;P(x=11\;or\;x=10)=P(11)+P(10)$

$\small P(x=11\;or\;x=10)=0.05+0.05$

$\small P(x=11\;or\;x=10)=\mathbf{{\color{red} 0.10}}$

--------------------------------------------------------------------------------------------------------------

$\small {\color{Blue} \mathbf{e)}}\;Mean=\sum x*P(x)$

$\small \mathbf{Mean=\mathbf{{\color{Red} 16.39}} }$

--------------------------------------------------------------------------------------------------------------

$\small {\color{Blue} \mathbf{f)}}\;Standard\;deviation=\sqrt{\sum x^{2}*P(x)-\left (\sum x*P(x) \right )^{2}}$

$\small Standard\;deviation=\sqrt{365.47 -\left (16.39 \right )^{2}}$

$\small Standard\;deviation=\mathbf{{\color{Red} 9.8406}}$