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 \(z = - 12\) and \(P(z)=0.22\): \((-12)\times0.22=-2.64\)
• For \(z=-11\) and \(P(z) = 0.06\): \((-11)\times0.06=-0.66\)
• For \(z=-10\) and \(P(z)=0.48\): \((-10)\times0.48=-4.8\)
• For \(z=-9\) and \(P(z)=0.17\): \((-9)\times0.17=-1.53\)
• For \(z=-8\) and \(P(z)=0.03\): \((-8)\times0.03=-0.24\)
• For \(z=-7\) and \(P(z)=0.04\): \((-7)\times0.04=-0.28\)

Step3: Sum up the terms

\(E(Z)=(-2.64)+(-0.66)+(-4.8)+(-1.53)+(-0.24)+(-0.28)\)

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Answer:

\(-10.15\)