Question

Calculate the standard deviation σ of X for the probability distribution. (Round your answer to two decimal places.)σ =x 1 2 3 4P(X = x)0.2 0.2 0.2 0.4Calculate the standard deviation σ of X for the probability distribution. (Round your answer to two decimal places.)σ =x −20 −10 0 10 20 30P(X = x)0.1 0.2 0.4 0.1 0 0.2

Answer 1

Answer:

a) $Sd(X)=\sqrt{Var(X)}=\sqrt{1.36}=1.166$

b) $Sd(X)=\sqrt{Var(X)}=\sqrt{149}=12.21$

Step-by-step explanation:

Part a

So then the random variable is given by this table

X     | 1      | 2    | 3    | 4     |

P(X) | 0.2 | 0.2 | 0.2 | 0.4 |

First we need to find the expected value (first moment) and the second moment in order to find the variance and then the standard deviation.

In order to calculate the expected value we can use the following formula:

$E(X)=\sum_{i=1}^n X_i P(X_i)$

And if we use the values obtained we got:

$E(X)=1*0.2 +2*0.2 +3*0.2 +4*0.4=2.8$

In order to find the standard deviation we need to find first the second moment, given by :

$E(X^2)=\sum_{i=1}^n X^2_i P(X_i)$

And using the formula we got:

$E(X^2)=(1^2 *0.2)+(2^2 *0.2)+(3^2 *0.2)+(4^2 *0.4)=9.2$

Then we can find the variance with the following formula:

$Var(X)=E(X^2)-[E(X)]^2 =9.2-(2.8)^2 =1.36$

And then the standard deviation would be given by:

$Sd(X)=\sqrt{Var(X)}=\sqrt{1.36}=1.17$

Part b

So then the random variable is given by this table

X     | -20  | -10 | 0   | 10  |20  |

P(X) | 0.1 | 0.2 | 0.4 | 0.1 | 0.2  |

First we need to find the expected value (first moment) and the second moment in order to find the variance and then the standard deviation.

In order to calculate the expected value we can use the following formula:

$E(X)=\sum_{i=1}^n X_i P(X_i)$

And if we use the values obtained we got:

$E(X)=(-20*0.1) +(-10*0.2) +(0*0.4) +(10*0.1)+(20*0.2)=1$

In order to find the standard deviation we need to find first the second moment, given by :

$E(X^2)=\sum_{i=1}^n X^2_i P(X_i)$

And using the formula we got:

$E(X^2)=((-20)^2 *0.1)+((-10)^2 *0.2)+(0^2 *0.4)+(10^2 *0.1)+(20^2 *0.2)=150$

Then we can find the variance with the following formula:

$Var(X)=E(X^2)-[E(X)]^2 =150-(1)^2 =149$

And then the standard deviation would be given by:

$Sd(X)=\sqrt{Var(X)}=\sqrt{149}=12.21$