Understanding Continuous and Discrete Distributions: Uniform, Normal, and Exponential

1. Section 7 – Continuous Distributions

2. Uniform • The probability of each X in the interval is “uniform” (the same) Uniform can be discrete (ex: dice only have integers) or continuous (all values in interval)

3. Uniform: Discrete vs. Continous Discrete Continuous

4. Normal Distribution • Notation: X ~ N(mean, variance) • Note: the second number is variance not standard deviation • Ex: Standard Normal, Z ~ N(0, 1) • You will likely not need to know the pdf • You will be given a normal table to find common z values • One-sided: use the (1 – alpha) percentile • Two-sided: use the (1 – alpha/2) percentile • P(r<X<s) = P[(r-µ)/σ < (X-µ)/σ < (s-µ)/σ ] • Be comfortable using the table!!!

5. Normal Approximation of Other Distribtutions • Given RV X, mean, and variance of the distribution (without knowing what the real distribution is) • Use normal distribution with the same mean/variance to approximate the true probability • Integer Correction for Discrete Distributions • P(n<=X<=m) becomes P(n-1/2<=X<=m+1/2) • Explained/justified well in Actex (see p. 200) • You’ll likely still be closest to the right answer without this, but it’s more accurate this way

6. When do I divide by sqrt(n)? • Estimating a value (X) • No square root of n • Estimating a mean (X bar) given a sample size • Involves square root of n • (ex: SOA 123 #81) • Hint: you only use n when given a sample size, and it’s used to decrease the size of the interval b/c an average is less variable • Note: the sample size does not affect the mean, only the variance

7. Exponential Distribution • Usage: X is time until an event occurs • Parameter: Lamba (mean = 1/Lambda) • Alternative: Use Theta = 1/Lambda • Theta = Mean • Can rewrite pdf, E[X], Var[X], etc. using Theta

8. “Memoryless” Property • Concept: what happened before doesn’t affect what’s going to happen now • Exponential is about time until an event occurs • Ex: if no insurance claims have happened in the past month, the exponential doesn’t think that one is “due” now • There is just as much chance of a claim happening this week as there was in the week following the first claim

9. Link Between Exponential and Poisson Distributions • These distributions are connected by the same parameter: Lambda • X is the time between events (continuous) • time per events • X ~ Exponential, with mean (1/Lambda) • Ex: time between claims • N is the number of events that have occurred while that time elapsed (discrete) • events per time • N ~ Poisson, with mean Lambda • Ex: number of claims in a period of time

10. Minimum of Multiple Exponential RV’s • Given multiple RV’s with exponential distributions and their means (1/Lambda), find some probability involving the minimum of all of the RV’s (the lowest value of all of the RV’s  the time at which the first event occurs) • An RV with a higher mean may still occur before the lower means (due to randomness) • Trap: do not just add the means of the exponential distributions • Technique • Convert to Poisson distributions, each with mean Lambda • Add the Lambda’s • This new Lambda is the parameter for Y = min{Y1, Y2, …, Yn} • Y ~ exponential with mean (1/Lambda) • Key Point: don’t add the means of the exponentials, convert to lambda’s, add the lambda’s, convert back to exponential • This is a harder problem, but VERY COMMONLY TESTED

11. Gamma Distribution • Actex does not recognize this as an important distribution to know • Uniform, Normal, and Exponential are the MOST important f(x) = βα*xα-1*e-βx Γ(α) Γ(n) = (n-1)! (if n is a positive integer) E(X) = α/β Var(X) = α/β2