29 Questions
What is the purpose of calculating the expected value in a probability distribution?
To quantify the average loss or gain of an event
What is the mathematical formula to calculate the expected value of a probability distribution?
Exp.Value = ∑(xi × pi)
What is the interpretation of a low variance in a probability distribution?
Low risk
What is the formula to calculate the variance of a probability distribution?
Variance = ∑(pi × (xi - μ)^2)
What is the calculated expected value of the given data?
1000
What is the relationship between the standard deviation and the variance of a probability distribution?
Standard deviation is the square root of the variance
What is the purpose of risk pooling in insurance companies?
To spread risk over a large group of individuals facing similar risks
What is the formula for calculating variance?
𝑁 𝑖=1 𝑝𝑖 (𝑥𝑖 − 𝜇)²
What is the first step in calculating the expected value of a probability distribution?
Multiply each possible outcome by its probability
What is the calculated standard deviation of the given data?
353.55
What is the primary advantage of pooling risks?
It creates a lower probability that an adverse situation will occur
What is the probability of no accident occurring in the example of Billy and Bully?
90%
What is the purpose of calculating the expected value?
To find the average outcome of an experiment
What is the concept of risk pooling based on?
The idea that the cost of individuals with higher risk is offset by those with lower risk
What is the formula for calculating the standard deviation?
√(Σ(xi - μ)²pi)
What is the significance of the correlation between the chance of losses for Billy and Bully?
It is uncorrelated
What is the purpose of calculating the expected loss, variance, and standard deviation?
To assess the risk of a particular situation
What is the probability of each outcome for a die tossed?
1/6
What is the maximum possible loss?
The maximum dollar amount of losses in the worst case scenario
What is the shape of a normal probability distribution?
A bell curve
What is skewness in a probability distribution?
The lack of asymmetry of a probability distribution
What is the law of large numbers?
When an experiment is performed a large number of times, the average results will approach the expected value
What is the expected value of a real-valued random variable?
The central distribution of the variable
What is the effect on the standard deviation when Belly joins the pooling arrangement?
It decreases
What is the value of the pooled loss (in RM) when there is an accident involving Billy and Belly?
15000
What is the probability of no accident for all three individuals?
0.729
What is the expected loss (in RM) when Belly joins the pooling arrangement?
1666.67
What is the variance of the pooled loss when Belly joins the arrangement?
8027759
What is the purpose of pooling the losses among Billy, Bully, and Belly?
To decrease the risk borne by each individual
Study Notes
Expected Value and Variance
- Expected value is the long-term average value of a random variable, calculated by multiplying each possible outcome by its probability and adding up the results.
- The mathematical function of the expected value can be expressed as:
Exp.Value = x1p1 + x2p2 + x3p3 +……+ xnpn = ∑(xi.pi). - Example: Calculate the expected value of a motorcycle accident with possible outcomes and probabilities:
Exp.Value = (0)(0.5) + (500)(0.3) + (1000)(0.1) + (2500)(0.05) + (5000)(0.05) + (10000)(0.02) = RM825.
Variance and Standard Deviation
- Variance is the spread of outcomes around the expected value, used to measure risk.
- Standard deviation is the square root of variance, commonly used to measure risk.
- Variance formula:
Variance = ∑(pi.(xi - μ)²). - Standard deviation formula:
Std Deviation = √(Variance). - Example: Calculate the variance and standard deviation of a loss with three possible outcomes:
RM500, RM1,000, and RM1,500with probabilities0.25, 0.50, and 0.25respectively.
Risk Pooling
- Risk pooling is used by insurance companies to spread risk over a large group of individuals facing similar risks.
- The concept of risk pooling: the cost of individuals with higher risk is offset by those with lower risk, creating a lower probability of an adverse situation occurring.
- Example: Two men, Billy and Bully, are exposed to the risk of accident, with a probability of 10% and a loss of
RM5,000. They decide to pool their risks, reducing the expected loss and variance.
Probability Distributions
- A normal probability distribution is shaped like a bell curve, with the expected value at the center.
- Skewness refers to the lack of asymmetry of a probability distribution, which can be negative (to the left) or positive (to the right).
- Characteristics of probability distributions include the expected value, variance, and standard deviation.
Pooling Risks and Expected Loss
- When individuals pool their risks, the expected loss and variance are reduced.
- The probability distribution of pooled risks changes, with the expected loss being the average of the individual losses.
- Example: Suppose another person, Belly, joins the pooling arrangement with Billy and Bully, the probability distribution will again change, and the expected loss and variance will be reduced.
Test your understanding of probability concepts with this quiz, featuring examples of equal and non-equal probability outcomes, including dice tossing and motorcycle accident losses. Calculate probabilities and identify correct outcomes.
Make Your Own Quizzes and Flashcards
Convert your notes into interactive study material.
Get started for free
