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Stat 446 Homework 2

1. An urn contains 10 red and 25 green balls. 6 balls are drawn without replacement. Determine the probability that there are 3 red and 3 green balls drawn.  (Hint: this is a hypergeometric distribution).

2. Drivers are classified as preferred, standard and high risk.  The annual number of claims for preferred drivers follows a Poisson distribution with mean of 0.15. The annual number of claims for standard drivers follows a Poisson distribution with mean of 0.35. The annual number of claims for high risk drivers follows a Poisson distribution with mean of 1.1.  The distribution of drivers is 20% preferred, 70% standard and 10% high risk.

a. A driver is selected at random. Determine the expected number of claims for the driver.

b. A driver is selected at random. Determine the probability that they had one claim in the year.

3. Suppose a random variable for claim frequency is Poisson and 40 claims are expected in one month. Claims are categorized as small, medium or large. The probability of a small claim is 0.6. The probability of a medium claim is 0.25. The probability of a large claim is 0.15. Determine the probability that there are less 3 medium claims in a month.

4. For a member of the (a,b,0) class of distributions, you are given that a = -4, b = 36 and p_k is the probability that X=k.

a. Determine what distribution is being used and it’s associated parameters.

b. Determine p₀, p₁ and p₂

5. For a member of the (a,b,0) class of distributions, you are given that and where p_k is the probability that X=k.

a. Determine what distribution is being used and it’s associated parameters.

b. Calculate the mean of the distribution.

c. Suppose that we are only interested the distribution of non-zero claims, find a recursive expression in terms of a and b for the Zero Truncated distribution of claims (i.e. find P_z^T(k)).

6. A zero modified distribution is in the (a,b,1) class and you are given the p₃ = 0.18737; p₄ = 0.06246; and p₅ = 0.18737.

a. Determine the values of α and β where α and β would take the roles of a and b in the truncated distribution.

b. Determine which distribution is being used and the values of the parameters.

c. Using the recursive relationship p_x =( α+β/x)*p_x‐1 determine the values of p₁ and p_2.

d. Determine the value of P(0).