Math Problem Statement
c 0 has a prob of 0.4, c 1 has a prob of 0.3, c 2 has a prob of 0.2, c 3 has a prob of 0.1, find the mean and standard deviation of c
Solution
To find the mean and standard deviation of the random variable with given probabilities, we need to follow these steps:
1. Calculate the Mean (Expected Value)
The mean is given by the formula:
Where represents the probability of each outcome .
Here are the values for and their corresponding probabilities:
- ,
- ,
- ,
- ,
So, the mean is calculated as:
2. Calculate the Variance
The variance is given by:
We already know that . Now we calculate the squared differences from the mean for each value of :
- For ,
- For ,
- For ,
- For ,
Now, use these values to calculate the variance:
3. Calculate the Standard Deviation
The standard deviation is the square root of the variance:
Final Answer:
- Mean
- Standard Deviation
Let me know if you'd like further details or have any questions!
Related Questions:
- How do you calculate the expected value for a discrete random variable?
- What is the formula for variance, and how is it different from the standard deviation?
- How would you calculate the mean and standard deviation for a continuous random variable?
- What does the standard deviation represent in terms of data spread?
- What would happen if the probabilities of the outcomes were different?
Tip: Always ensure the sum of the probabilities for a random variable equals 1. If it doesn’t, check the given probabilities for errors.
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Math Problem Analysis
Mathematical Concepts
Probability
Random Variables
Statistics
Formulas
Mean (Expected Value) \( \mu = \sum_{i} P(C_i) \cdot C_i \)
Variance \( \sigma^2 = \sum_{i} P(C_i) \cdot (C_i - \mu)^2 \)
Standard Deviation \( \sigma = \sqrt{\sigma^2} \)
Theorems
Law of Total Expectation
Properties of Variance
Suitable Grade Level
Grades 9-12
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