Find the mean and standard deviation for the random variable x given the following distribution

Question

Find the mean and standard deviation for the random variable x given the following distribution

X                    4          5                   6                    7                8

P(X=x)         0.10       0.30         0.45                   0.10              0.05

Answer 1

The first thing you do is to rearrange the numbers.

Then, multiply the 1st column by the 2nd column to get the 3rd column

Next, you add up the 3rd column to get the mean.

X              P(X = x)         (x * Px)    x - µ        (x - µ) ^2        (x- µ) ^ 2* p(X = x)

4                  0.1                0.4       4 - 5.7      2.89                         0.289

5                  0.3                1.5       5 - 5.7      0.49                         0.147

6                  0.45              2.7       6 - 5.7       0.09                        0.0405

7                  0.10              0.7       7 - 5.7       1.69                        0.169

8                  0.05              0.4       8 - 5.7       5.29                        0.2645

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

                            Total = 5.7 (mean, µ)                          Total = 0.91 (variance)

To calculate the standard deviation, develop columns 4, 5, and 6.

Finally, you sum up column 6 to arrive at the variance.

 

Remember that Standard Deviation is the Square root of Variance

Therefore, square root of 0.91 = 0.95 (std. dev.)

If you abide by the above methodology, the sky is definitely the limit for you. Thank you.

                         

Answer 2

There is yet another approach to finding the mean and standard deviation of this distribution. As a probability problem, the mean is the same as the expected value. Expected value is calculated as the products of all outcomes and their probabilities, all added together.

So the mean E(X) = 4*0.1 + 5*0.3 + 6*0.45 + 7*0.1 + 8*0.05 = 5.7.

The standard deviation can be calculated another way, by first finding the variance using this formula: Var(X) = E(X²) - [E(X)]².

For E(X²), we're multiplying the squares of the outcomes by their probabilities, and taking their sum.

E(X²) = 16*0.1 + 25*0.3 + 36*0.45 + 49*0.1 + 64*0.05 = 33.4.

So Var(X) = 33.4 - 5.7² = 0.91.

The standard deviation is the square root of the variance, so σ = 0.95.

There is a way to prove this equivalency, but it is long, messy, and tricky.

Note: This formula only works for population standard deviation.

Answer 3

There are quicker ways to solve this problem than the method I will show you, but I think doing it the long hand way will help you understand it better. The distribution given says that 10% of the time the random variable x is 4, 30% of the time x is 5, etc. Since the smallest percentage in the distribution is 5% (when x equals 8) we will assume a population size of 20 so that each x value in the population is equivalent to 5% of the distribution. When using 20 numbers the distribution would be: 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 8 (two 4s, six 5s, nine 6s, two 7s, and one 8).

The mean is equal to [(2)*(4)+(6)*(5)+(9)*(6)+(2)*(7)+(1)*(8)] /20 = 5.7

You will get the same answer regardless of the population size you choose. You can prove this to yourself if you choose a population size of 1.

The mean would when be calculated as [(0.1)*(4)+(0.3)*(5)+(0.45)*(6)+(0.1)*(7)+(0.05)*(8)]/1=5.7

The standard deviation to three decimal places is equal to 0.954. The calculation is a bit long to show here, but your textbook should show you how to calculate the standard deviation for a set of numbers.  Make sure to use the formula for a population, not a sample!  I think you may know how to do this step, but were having difficulty figuring out the distribution portion of the question. Complete your calculation and check the result.