SOLUTION: A random variable x has the following probability mass function defined in tabular form as shown below x values are -1,1,2 and p(x=x) 2c,3c ,4c
Find the value of c and computer p(
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Question 1202695: A random variable x has the following probability mass function defined in tabular form as shown below x values are -1,1,2 and p(x=x) 2c,3c ,4c
Find the value of c and computer p(x=-1),find the standard deviation of x
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!
To have a valid probability distribution, the p(x) values must add to 1.
2c+3c+4c = 1
9c = 1
c = 1/9
So,
2c = 2*(1/9) = 2/9
3c = 3*(1/9) = 3/9 = 1/3
4c = 4*(1/9) = 4/9
We get this updated table.
We see that P(x=-1) = 2/9
2/9 = 0.222 approximately
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This video goes over an example how to calculate the standard deviation of a probability distribution.
https://www.youtube.com/watch?v=YMbt5rYzp-Q
The professor mentions these two formulas
and
where,
mu = = greek letter to represent the mean
lowercase sigma = = greek letter for the standard deviation
The fancy looking "E" represents a summation. It's the greek uppercase letter sigma.
Something like tells us to add up all of the x^2*P(x) terms.
Both formulas involve mu, so let's determine that value first.
mu =
mu = add up the x*P(x) terms
| x | P(x) | x*P(x) |
| -1 | 2/9 | -2/9 |
| 1 | 1/3 | 1/3 |
| 2 | 4/9 | 8/9 |
The sum of the x*P(x) values is:
-2/9 + 1/3 + 8/9
= -2/9 + 3/9 + 8/9
= (-2 + 3 + 8)/9
= 9/9
= 1
Therefore, mu = 1.
Let's say we used the formula
What we'll need to do is- Subtract mu from each x value to get x-mu.
- Square the difference to get (x-mu)^2
- Multiply those squares with P(x) to get (x-mu)^2*P(x).
This is what your table should look like
| x | P(x) | x-mu | (x-mu)^2 | (x-mu)^2*P(x) |
| -1 | 2/9 | -2 | 4 | 8/9 |
| 1 | 1/3 | 0 | 0 | 0 |
| 2 | 4/9 | 1 | 1 | 4/9 |
Add up the values in the (x-mu)^2*P(x) column.
8/9 + 0 + 4/9 = 12/9 = 4/3
That's the variance, so the standard deviation is the square root of that
The standard deviation is exactly (2/3)*sqrt(3)
When using a calculator, (2/3)*sqrt(3) = 1.15470053837926 approximately.
If you wanted to use the formula , then this is what your table could look like
| x | P(x) | x*P(x) | x^2 | x^2*P(x) |
| -1 | 2/9 | -2/9 | 1 | 2/9 |
| 1 | 1/3 | 1/3 | 1 | 1/3 |
| 2 | 4/9 | 8/9 | 4 | 16/9 |
Add up the x^2*P(x) values
Then plug that result into the formula I mentioned earlier
This calculator can be used to check your work
https://www.mathportal.org/calculators/statistics-calculator/probability-distributions-calculator.php
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Summary:
c = 1/9
P(x = -1) = 2/9
Standard deviation = (2/3)*sqrt(3) = 1.15470053837926 approximately
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