Question

the table below shows the probability distribution of a random variable z.
what is the expected value of z?
write your answer as a decimal.

Explanation:

Step1: Recall the formula for expected value

The formula for the expected value \( E(Z)\) of a discrete random variable is \( E(Z)=\sum_{i}z_{i}P(z_{i})\), where \( z_{i}\) are the values of the random variable and \( P(z_{i})\) are their corresponding probabilities.

Step2: Substitute the values into the formula

For \( z = 7\) and \( P(7)=0.2\), the contribution is \( 7\times0.2\). For \( z = 8\) and \( P(8)=0.6\), the contribution is \( 8\times0.6\). For \( z = 9\) and \( P(9)=0.2\), the contribution is \( 9\times0.2\).
So, \( E(Z)=(7\times0.2)+(8\times0.6)+(9\times0.2)\)

Step3: Calculate each term

\( 7\times0.2 = 1.4\), \( 8\times0.6=4.8\), \( 9\times0.2 = 1.8\)

Step4: Sum the terms

\( E(Z)=1.4 + 4.8+1.8\)
\( E(Z)=8.0\)

Answer:

\(8.0\)