CFA Level 1 - Quantitative Methods & Economics

The Mean is the arithmetic average of a dataset. The Median is the middle value when data is ordered. The Mode is the most frequently occurring value in a dataset.

Variance (${\sigma }^2$) measures the average of the squared differences from the mean, indicating data dispersion. Standard Deviation ($\sigma$) is the square root of variance, providing a measure of dispersion in the original units of the data.

Skewness describes the asymmetry of a distribution. Positive skew means a long tail to the right; negative skew means a long tail to the left. Kurtosis measures the 'tailedness' or peakedness of a distribution. Leptokurtic (high kurtosis) means fatter tails and a sharper peak than a normal distribution; platykurtic (low kurtosis) means thinner tails and a flatter peak.

Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as $P(A|B)$, which is the probability of event A given event B, calculated as $P(A|B)=P(A\cap B)/P(B)$.

Bayes' Theorem updates the probability of an event based on new information. It is given by $P(A|B)=[P(B|A)\ast P(A)]/P(B)$. It's used to calculate a posterior probability.

A normal distribution is bell-shaped and symmetric around its mean. Its mean, median, and mode are equal. Approximately 68% of observations fall within $\pm 1$ standard deviation, 95% within $\pm 2$ standard deviations, and 99% within $\pm 3$ standard deviations of the mean.

The Central Limit Theorem states that for a sufficiently large sample size (typically $n\ge 30$), the sampling distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution. Its mean will be the population mean, and its standard deviation (standard error) will be $\sigma /\sqrt{n}$.

A Type I Error occurs when the null hypothesis ($H_0$) is rejected when it is actually true (false positive). Its probability is denoted by $\alpha$ (significance level). A Type II Error occurs when the null hypothesis ($H_0$) is not rejected when it is actually false (false negative). Its probability is denoted by eta.