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 \(z = - 5\) and \(P(-5)=0.27\): \((-5)\times0.27=-1.35\)
• For \(z=-4\) and \(P(-4) = 0.09\): \((-4)\times0.09=-0.36\)
• For \(z=-3\) and \(P(-3)=0.36\): \((-3)\times0.36=-1.08\)
• For \(z=-2\) and \(P(-2)=0.16\): \((-2)\times0.16=-0.32\)
• For \(z=-1\) and \(P(-1)=0.12\): \((-1)\times0.12=-0.12\)

Step3: Sum up the terms

\(E(Z)=-1.35+( - 0.36)+(-1.08)+(-0.32)+(-0.12)\)

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Answer:

\(-3.23\)