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Math  /  Data & Statistics

QuestionLet XX be a random variable with the following probability distribution \begin{tabular}{|c|c|} \hline Value xx of XX & P(X=x)P(X=x) \\ \hline 30 & 0.05 \\ \hline 40 & 0.05 \\ \hline 50 & 0.15 \\ \hline 60 & 0.15 \\ \hline 70 & 0.35 \\ \hline 80 & 0.25 \\ \hline \end{tabular}
Complete the following. (If necessary, consult a list of formulas.) (a) Find the expectation E(X)E(X) of XX. E(X)=E(X)= □\square

Studdy Solution

STEP 1

1. The random variable XX has a discrete probability distribution given by the specified table.
2. The expectation (or mean) E(X)E(X) of a discrete random variable XX can be calculated using the formula: E(X) = \sum_{i} x_i P(X = x_i) \] where x_iarethepossiblevaluesof are the possible values of Xand and P(X = x_i)$ are the corresponding probabilities.

STEP 2

1. Identify and list the values xix_i of XX and their corresponding probabilities P(X=xi)P(X = x_i).
2. Apply the expectation formula to calculate E(X)E(X).
3. Sum the products of each value xix_i and its probability P(X=xi)P(X = x_i).

STEP 3

List the values xix_i of XX and their corresponding probabilities P(X=xi)P(X = x_i).
Value xiP(X=xi)300.05400.05500.15600.15700.35800.25\begin{array}{|c|c|} \hline \text{Value } x_i & P(X = x_i) \\ \hline 30 & 0.05 \\ \hline 40 & 0.05 \\ \hline 50 & 0.15 \\ \hline 60 & 0.15 \\ \hline 70 & 0.35 \\ \hline 80 & 0.25 \\ \hline \end{array}

STEP 4

Apply the expectation formula to calculate E(X)E(X).
E(X)=∑ixiP(X=xi)=30⋅0.05+40⋅0.05+50⋅0.15+60⋅0.15+70⋅0.35+80⋅0.25E(X) = \sum_{i} x_i P(X = x_i) = 30 \cdot 0.05 + 40 \cdot 0.05 + 50 \cdot 0.15 + 60 \cdot 0.15 + 70 \cdot 0.35 + 80 \cdot 0.25

STEP 5

Calculate the individual products.
30⋅0.05=1.540⋅0.05=2.050⋅0.15=7.560⋅0.15=9.070⋅0.35=24.580⋅0.25=20.0\begin{aligned} 30 \cdot 0.05 &= 1.5 \\ 40 \cdot 0.05 &= 2.0 \\ 50 \cdot 0.15 &= 7.5 \\ 60 \cdot 0.15 &= 9.0 \\ 70 \cdot 0.35 &= 24.5 \\ 80 \cdot 0.25 &= 20.0 \\ \end{aligned}

STEP 6

Sum the products to find E(X)E(X).
E(X)=1.5+2.0+7.5+9.0+24.5+20.0=64.5E(X) = 1.5 + 2.0 + 7.5 + 9.0 + 24.5 + 20.0 = 64.5
Solution: E(X)=64.5E(X) = 64.5

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