Question

the table shows claims and their probabilities for an insurance company.
amount of claim (to the nearest $20,000) | probability
$0 | 0.70
$20,000 | 0.18
$40,000 | 0.07
$60,000 | 0.03
$80,000 | 0.01
$100,000 | 0.01

a. calculate the expected value.
$

Explanation:

Step1: Recall the formula for expected value

The expected value \( E(X) \) of a discrete random variable is calculated as \( E(X)=\sum_{i} x_i P(x_i) \), where \( x_i \) is the value of the random variable and \( P(x_i) \) is the corresponding probability.

Step2: Calculate each term \( x_i P(x_i) \)

• For \( x_1 = 0 \) and \( P(x_1)=0.70 \): \( 0\times0.70 = 0 \)
• For \( x_2 = 20000 \) and \( P(x_2)=0.18 \): \( 20000\times0.18=3600 \)
• For \( x_3 = 40000 \) and \( P(x_3)=0.07 \): \( 40000\times0.07 = 2800 \)
• For \( x_4 = 60000 \) and \( P(x_4)=0.03 \): \( 60000\times0.03=1800 \)
• For \( x_5 = 80000 \) and \( P(x_5)=0.01 \): \( 80000\times0.01 = 800 \)
• For \( x_6 = 100000 \) and \( P(x_6)=0.01 \): \( 100000\times0.01=1000 \)

Step3: Sum all the terms

\( E(X)=0 + 3600+2800 + 1800+800+1000 \)
\( E(X)=3600 + 2800=6400 \); \( 6400+1800 = 8200 \); \( 8200+800=9000 \); \( 9000 + 1000=10000 \)

Answer:

\( 10000 \)