Following are three economic states, their likelihoods, and the potential returns: Economic State Probability Return Fast growth 0.3 40 % Slow growth 0.4 10 Recession 0.3 –25 Determine the standard deviation of the expected return. (Do not round intermediate calculations and round your answer to 2 decimal places.)

In statistics and probability analysis, the expected value "is calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur and then summing all of those values".

The variance of a random variable Var(X) is the expected value of the squared deviation from the mean of X, E(X).

And the standard deviation of a random variable X is just the square root of the variance.

The data given is:

Economic Sate Probability Return

Fast Growth 0.3 40%

Slow Growth 0.4 10%

Rcession 0.3 25%

So then the random variable is given by this table

X | 40 | 10 | 25 |

P(X) | 0.3 | 0.4 | 0.3 |

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

[tex]E(X)=\sum_{i=1}^n X_i P(X_i)[/tex]

And if we use the values obtained we got:

[tex]E(X)=40*0.3 + 10*0.4 + 25*0.3=23.5[/tex]

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

[tex]E(X^2)=\sum_{i=1}^n X^2_i P(X_i)[/tex]

And using the formula we got:

[tex]E(X^2)=40^2*0.3 + 10^2*0.4 + 25^2*0.3=707.5[/tex]

Then we can find the variance with the following formula:

[tex]Var(X)=E(X^2)-[E(X)]^2 =707.5-(23.5)^2 =155.25[/tex]

And then the standard deviation would be given by:

[tex]Sd(X)=\sqrt{Var(X)}=\sqrt{155.25}=12.46[/tex]