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 CC with given probabilities, we need to follow these steps:

1. Calculate the Mean (Expected Value) μ\mu

The mean μ\mu is given by the formula:

μ=E(C)=iP(Ci)Ci\mu = E(C) = \sum_{i} P(C_i) \cdot C_i

Where P(Ci)P(C_i) represents the probability of each outcome CiC_i.

Here are the values for CiC_i and their corresponding probabilities:

  • C0=0C_0 = 0, P(C0)=0.4P(C_0) = 0.4
  • C1=1C_1 = 1, P(C1)=0.3P(C_1) = 0.3
  • C2=2C_2 = 2, P(C2)=0.2P(C_2) = 0.2
  • C3=3C_3 = 3, P(C3)=0.1P(C_3) = 0.1

So, the mean is calculated as:

μ=(0×0.4)+(1×0.3)+(2×0.2)+(3×0.1)\mu = (0 \times 0.4) + (1 \times 0.3) + (2 \times 0.2) + (3 \times 0.1) μ=0+0.3+0.4+0.3=1.0\mu = 0 + 0.3 + 0.4 + 0.3 = 1.0

2. Calculate the Variance

The variance σ2\sigma^2 is given by:

σ2=iP(Ci)(Ciμ)2\sigma^2 = \sum_{i} P(C_i) \cdot (C_i - \mu)^2

We already know that μ=1\mu = 1. Now we calculate the squared differences from the mean for each value of CiC_i:

  • For C0=0C_0 = 0, (01)2=1(0 - 1)^2 = 1
  • For C1=1C_1 = 1, (11)2=0(1 - 1)^2 = 0
  • For C2=2C_2 = 2, (21)2=1(2 - 1)^2 = 1
  • For C3=3C_3 = 3, (31)2=4(3 - 1)^2 = 4

Now, use these values to calculate the variance:

σ2=(0.4×1)+(0.3×0)+(0.2×1)+(0.1×4)\sigma^2 = (0.4 \times 1) + (0.3 \times 0) + (0.2 \times 1) + (0.1 \times 4) σ2=0.4+0+0.2+0.4=1.0\sigma^2 = 0.4 + 0 + 0.2 + 0.4 = 1.0

3. Calculate the Standard Deviation

The standard deviation σ\sigma is the square root of the variance:

σ=σ2=1.0=1.0\sigma = \sqrt{\sigma^2} = \sqrt{1.0} = 1.0

Final Answer:

  • Mean μ=1.0\mu = 1.0
  • Standard Deviation σ=1.0\sigma = 1.0

Let me know if you'd like further details or have any questions!

Related Questions:

  1. How do you calculate the expected value for a discrete random variable?
  2. What is the formula for variance, and how is it different from the standard deviation?
  3. How would you calculate the mean and standard deviation for a continuous random variable?
  4. What does the standard deviation represent in terms of data spread?
  5. What would happen if the probabilities of the outcomes C0,C1,C2,C3C_0, C_1, C_2, C_3 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