Answer :
To find the expected value [tex]\(\mu\)[/tex] of the random variable given the probability distribution, follow these steps:
1. Understand the definition of expected value: The expected value (or mean) of a discrete random variable is the sum of the products of each value of the random variable and its corresponding probability.
2. Set up the formula: If [tex]\(X\)[/tex] is a random variable with values [tex]\(x\)[/tex] and corresponding probabilities [tex]\(P(x)\)[/tex], the expected value [tex]\(\mu\)[/tex] is calculated as:
[tex]\[ \mu = \sum_{i} x_i P(x_i) \][/tex]
where [tex]\(x_i\)[/tex] are the values of the random variable, and [tex]\(P(x_i)\)[/tex] are the corresponding probabilities.
3. List the values and probabilities: From the given table, we have:
[tex]\[ \begin{array}{c|c} x & P(x) \\ \hline 1 & 0.11 \\ 3 & 0.19 \\ 5 & 0.26 \\ 7 & 0.21 \\ 9 & 0.14 \\ 11 & 0.09 \\ \end{array} \][/tex]
4. Multiply each value by its probability and sum the products:
[tex]\[ \begin{align*} 1 \times 0.11 &= 0.11, \\ 3 \times 0.19 &= 0.57, \\ 5 \times 0.26 &= 1.30, \\ 7 \times 0.21 &= 1.47, \\ 9 \times 0.14 &= 1.26, \\ 11 \times 0.09 &= 0.99. \end{align*} \][/tex]
5. Sum up all the products:
[tex]\[ 0.11 + 0.57 + 1.30 + 1.47 + 1.26 + 0.99 = 5.7 \][/tex]
Therefore, the expected value [tex]\(\mu\)[/tex] of the random variable is:
[tex]\[ \mu = 5.7 \][/tex]
1. Understand the definition of expected value: The expected value (or mean) of a discrete random variable is the sum of the products of each value of the random variable and its corresponding probability.
2. Set up the formula: If [tex]\(X\)[/tex] is a random variable with values [tex]\(x\)[/tex] and corresponding probabilities [tex]\(P(x)\)[/tex], the expected value [tex]\(\mu\)[/tex] is calculated as:
[tex]\[ \mu = \sum_{i} x_i P(x_i) \][/tex]
where [tex]\(x_i\)[/tex] are the values of the random variable, and [tex]\(P(x_i)\)[/tex] are the corresponding probabilities.
3. List the values and probabilities: From the given table, we have:
[tex]\[ \begin{array}{c|c} x & P(x) \\ \hline 1 & 0.11 \\ 3 & 0.19 \\ 5 & 0.26 \\ 7 & 0.21 \\ 9 & 0.14 \\ 11 & 0.09 \\ \end{array} \][/tex]
4. Multiply each value by its probability and sum the products:
[tex]\[ \begin{align*} 1 \times 0.11 &= 0.11, \\ 3 \times 0.19 &= 0.57, \\ 5 \times 0.26 &= 1.30, \\ 7 \times 0.21 &= 1.47, \\ 9 \times 0.14 &= 1.26, \\ 11 \times 0.09 &= 0.99. \end{align*} \][/tex]
5. Sum up all the products:
[tex]\[ 0.11 + 0.57 + 1.30 + 1.47 + 1.26 + 0.99 = 5.7 \][/tex]
Therefore, the expected value [tex]\(\mu\)[/tex] of the random variable is:
[tex]\[ \mu = 5.7 \][/tex]
