Step 1: First of all we will find out Probabilities for each and every Random Variable means we will calculate Probability Distribution for that particular random variable like, if ‘X’ is a random variable, then probability distribution of ‘X’ is

X p(X)

x

_{1}0.7

x

_{2}0.9

x

_{3}0.4

x

_{4}0.2

Here p(X) is a probability value for each and every random variable X(x

_{1}, x

_{2}, x

_{3}, x

_{4}).

Step 2: After evaluation of probability distribution of the given random variable, we will calculate Mean value for that random variable which is called as an expected value of random variable by following method -

E(X) = x

_{1}* p(x

_{1}) + x

_{2}* p(x

_{2}) + x

_{3}* p(x

_{3}) + x

_{4}* p(x

_{4}),

Here x

_{1}, x

_{2}, x

_{3}, and x

_{4}are random variables and p(x

_{1}), p(x

_{2}), p(x

_{3}) and p(x

_{4}) are different probabilities for the random variable ‘X’.

Step 3: After evaluation of mean value or expected value for random variable, we calculate Square of this mean or expected value for the given probability distribution means we have to evaluate E(X

^{2}), where

E(X

^{2}) = (x

_{1}

^{2}) * p(x

_{1}) + (x

_{1}

^{2}) * p(x

_{1}) + (x

_{1}

^{2}) * p(x

_{1}) + (x

_{1}

^{2}) * p(x

_{1}),

Step 4: After evaluation of square of expected value for random variable, we calculate the variance for the given probability distribution by following method -

Variance for the given probability distribution = E(X

^{2}) – (E(X)

^{2}),

Step 5: Now we calculate standard deviation of given probability distribution by using the variance:

Standard deviation = √ (variance)

Suppose we have a given probability distribution -

X p(X)

0 5

1 7

2 4

3 3

We will now use above steps for find the standard deviation for the given probability distribution-

Step 1: First of all, we evaluate expected value -

E(X) = 0 * p(0) + 1 * p(1) + 2 * p(2) + 3 * p(3),

= 0 * 5 + 1 * 7 + 2 * 4 + 3 * 3,

= 0 + 7 + 8 + 9,

= 24.

Step 2: Now we evaluate variance of given probability distribution-

Variance = E(X

^{2}) – (E(X)

^{2})

= 696 – 24

^{2},

= 696 – 596,

= 100.

Step 3: Now we will evaluate standard deviation of given probability distribution -

Standard deviation = √ (variance),

= √ (100),

= 10,

So, variance of given probability distribution is 10.

Use the below widget to find standard deviation.

Use the below widget to find standard deviation.