To find the values of p and q, we use the following conditions:
- The sum of all probabilities must equal 1:
\(0.15 + p + 0.4 + q = 1\).
- The expected value is given by
\(E(X) = 3.05\), which means
\(1 \times 0.15 + 2 \times p + 3 \times 0.4 + 6 \times q = 3.05\).
From the first condition, we have:
\(p + q = 0.45\).
From the second condition, we have:
\(0.15 + 2p + 1.2 + 6q = 3.05\).
Simplifying the second equation:
\(2p + 6q = 3.05 - 0.15 - 1.2 = 1.7\).
Now, solve the simultaneous equations:
- \(p + q = 0.45\)
- \(2p + 6q = 1.7\)
From equation (1), express \(p\) in terms of \(q\):
\(p = 0.45 - q\).
Substitute into equation (2):
\(2(0.45 - q) + 6q = 1.7\).
\(0.9 - 2q + 6q = 1.7\).
\(0.9 + 4q = 1.7\).
\(4q = 1.7 - 0.9 = 0.8\).
\(q = 0.2\).
Substitute \(q = 0.2\) back into \(p = 0.45 - q\):
\(p = 0.45 - 0.2 = 0.25\).
Thus, the values are \(p = 0.25\) and \(q = 0.2\).
