Probability Scores
0.2 0
0.2 2
0.05 4
0.45 7
0.1 9
Find the variance of the above random variable random variable.
To find the variance of a random variable, we need to calculate the weighted average of the squared deviations from the mean.
First, we need to calculate the mean of the random variable. The mean (μ) can be calculated by multiplying each score by its respective probability and summing them up.
μ = (0 * 0.2) + (2 * 0.2) + (4 * 0.05) + (7 * 0.45) + (9 * 0.1)
μ = 0 + 0.4 + 0.2 + 3.15 + 0.9
μ = 4.65
Next, for each score, we subtract the mean and square the result, and then multiply it by its respective probability.
(0 - 4.65)^2 * 0.2 = 4.65^2 * 0.2 = 21.6225 * 0.2 = 4.3245
(2 - 4.65)^2 * 0.2 = 2.65^2 * 0.2 = 7.0225 * 0.2 = 1.4045
(4 - 4.65)^2 * 0.05 = 0.65^2 * 0.05 = 0.4225 * 0.05 = 0.021125
(7 - 4.65)^2 * 0.45 = 2.35^2 * 0.45 = 5.5225 * 0.45 = 2.484125
(9 - 4.65)^2 * 0.1 = 4.35^2 * 0.1 = 18.9225 * 0.1 = 1.89225
Finally, we sum up the weighted squared deviations:
Variance = 4.3245 + 1.4045 + 0.021125 + 2.484125 + 1.89225
Variance = 10.1265
Therefore, the variance of the given random variable is 10.1265.
To find the variance of a random variable, we need to follow these steps:
Step 1: Calculate the expected value (mean) of the random variable.
Step 2: Calculate the squared difference between each score and the expected value.
Step 3: Multiply each squared difference by its corresponding probability.
Step 4: Sum up all the values obtained in Step 3.
Step 5: The result from Step 4 is the variance.
Let's calculate the variance of the given random variable:
Step 1: Calculate the expected value (mean):
The expected value (mean) is calculated by multiplying each score by its corresponding probability and summing them up:
Expected value = (0 * 0.2) + (2 * 0.2) + (4 * 0.05) + (7 * 0.45) + (9 * 0.1)
= 0 + 0.4 + 0.2 + 3.15 + 0.9
= 4.65
So, the expected value (mean) of the random variable is 4.65.
Step 2: Calculate the squared difference between each score and the expected value:
For each score, subtract the expected value and then square the difference:
Squared difference for 0: (0 - 4.65)^2 = 21.6225
Squared difference for 2: (2 - 4.65)^2 = 6.9025
Squared difference for 4: (4 - 4.65)^2 = 0.4225
Squared difference for 7: (7 - 4.65)^2 = 5.6225
Squared difference for 9: (9 - 4.65)^2 = 26.7225
Step 3: Multiply each squared difference by its corresponding probability:
Multiply each squared difference obtained in Step 2 by its corresponding probability:
Probability Squared Difference Product
0.2 21.6225 4.3245
0.2 6.9025 1.3805
0.05 0.4225 0.0211
0.45 5.6225 2.5306
0.1 26.7225 2.6723
Step 4: Sum up all the products obtained in Step 3:
Sum up all the products obtained in Step 3 to get the variance:
Variance = Sum of Products
= 4.3245 + 1.3805 + 0.0211 + 2.5306 + 2.6723
= 10.929
So, the variance of the given random variable is 10.929.