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 \(E(Z)=\sum_{i}z_iP(z_i)\), where \(z_i\) are the values of the random variable and \(P(z_i)\) are their corresponding probabilities.

Step2: Calculate each term \(z_iP(z_i)\)

• For \(z = - 19\) and \(P(-19)=0.39\), the term is \((-19)\times0.39=-7.41\)
• For \(z=-18\) and \(P(-18) = 0\), the term is \((-18)\times0 = 0\)
• For \(z=-17\) and \(P(-17)=0.07\), the term is \((-17)\times0.07=-1.19\)
• For \(z=-16\) and \(P(-16)=0.32\), the term is \((-16)\times0.32=-5.12\)
• For \(z=-15\) and \(P(-15)=0.12\), the term is \((-15)\times0.12=-1.8\)
• For \(z=-14\) and \(P(-14)=0.04\), the term is \((-14)\times0.04=-0.56\)
• For \(z=-13\) and \(P(-13)=0.06\), the term is \((-13)\times0.06=-0.78\)

Step3: Sum up all the terms

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Answer:

\(-16.86\)