Question

If Return (R) in Percentage is 1,2,3 with the probability 1/2, 1/3,1/6 respectively. Then Variance Return is​

Answer 1

Step-by-step explanation:

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Answer 2

The Variance of the probability distribution [X; P(X)] 1;1/2, 2;1/3, 3;1/6 is  14/9.

Given:

• Probability distribution

       X    ;     P(X)

       1     ;      $\frac{1}{2}$

       2    ;      $\frac{1}{3}$

       3    ;      $\frac{1}{6}$

To Find:

The Variance

Solution:

A variance of a dataset describes the spread of all the individual values in the dataset. It can be mathematically summarised as:

• Var(X) = E(X²) - E(X)² , where

E(X) refers to the Expectation of the data set.

• In a probability distribution table E(X) of the dataset would be:

E(X) = ∑ XP(X),

summation of the product of the frequency of Event X and Probability of Event X.

Similarly E(X²) = ∑ X²P(X)

So, For finding Var(X)

 ⇒ E(X)² = ∑ XP(X)

               = [ (1 * $\frac{1}{2}$) + (2 * $\frac{1}{3}$) + (3 * $\frac{1}{6}$) ]²

               = ( $\frac{1}{2}$ + $\frac{2}{3}$ + $\frac{1}{2}$ )²

               = ( $\frac{4}{3}$ )² = $\frac{16}{9}$

 ⇒ E(X²) = ∑ X²P(X)

               =  (1² *  $\frac{1}{2}$) + (2² * $\frac{1}{3}$) + (3² * $\frac{1}{6}$)

               =  (1 *  $\frac{1}{2}$) + (4 * $\frac{1}{3}$) + (9 * $\frac{1}{6}$)

               =  ( $\frac{1}{2}$ + $\frac{4}{3}$ + $\frac{9}{6}$ )

               =  ( $\frac{10}{3}$ )

Var(X) = E(X²) - E(X)²

          =  $\frac{10}{3}$ - $\frac{16}{9}$

          = $\frac{14}{9}$

Hence, the variance is 14/9.

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