Math Problem Statement

If the random variable X denotes the real valued outcomes given in the following table with the given frequency of occurrence of each of the outcomes, fill in the last line of the table and find the expected value of X. Outcome -2 -1 0 1 2 Frequency 26 22 27 25 12 P(X)

E(X) =

Solution

To fill in the last line of the table, we need to find the probability P(X)P(X) for each outcome. The probability is calculated by dividing the frequency of each outcome by the total number of observations.

Step 1: Calculate the Total Frequency

Total Frequency=26+22+27+25+12=112\text{Total Frequency} = 26 + 22 + 27 + 25 + 12 = 112

Step 2: Find P(X)P(X) for Each Outcome

P(X=2)=26112=13560.2321P(X = -2) = \frac{26}{112} = \frac{13}{56} \approx 0.2321 P(X=1)=22112=11560.1964P(X = -1) = \frac{22}{112} = \frac{11}{56} \approx 0.1964 P(X=0)=27112=271120.2411P(X = 0) = \frac{27}{112} = \frac{27}{112} \approx 0.2411 P(X=1)=25112=251120.2232P(X = 1) = \frac{25}{112} = \frac{25}{112} \approx 0.2232 P(X=2)=12112=3280.1071P(X = 2) = \frac{12}{112} = \frac{3}{28} \approx 0.1071

The completed table with probabilities is:

Outcome-2-1012
Frequency2622272512
P(X)P(X)0.23210.19640.24110.22320.1071

Step 3: Calculate the Expected Value E(X)E(X)

E(X)=(xP(x))E(X) = \sum (x \cdot P(x)) E(X)=(20.2321)+(10.1964)+(00.2411)+(10.2232)+(20.1071)E(X) = (-2 \cdot 0.2321) + (-1 \cdot 0.1964) + (0 \cdot 0.2411) + (1 \cdot 0.2232) + (2 \cdot 0.1071) E(X)=0.46420.1964+0+0.2232+0.2142E(X) = -0.4642 - 0.1964 + 0 + 0.2232 + 0.2142 E(X)=0.2232E(X) = -0.2232

Therefore, the expected value of XX is: E(X)=0.2232E(X) = -0.2232

Would you like a more detailed explanation or have any questions?

Here are 5 related questions you might find interesting:

  1. How do you calculate variance and standard deviation for a discrete random variable?
  2. What would be the expected value if the outcomes had different weights?
  3. How does the expected value change if the frequency distribution is modified?
  4. Can the expected value be a non-integer even if all outcomes are integers?
  5. How can we use expected value in real-life situations, such as in gambling or insurance?

Tip: When finding probabilities, always ensure the sum of P(X)P(X) values equals 1, as a quick way to check your calculations.

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Math Problem Analysis

Mathematical Concepts

Probability
Statistics
Expected Value of a Discrete Random Variable

Formulas

P(X = x) = Frequency of outcome / Total frequency
E(X) = Σ(x * P(x))

Theorems

Law of Total Expectation

Suitable Grade Level

Grades 9-12