# Month: November 2018

### Practice Problem Set 11 – (a,b,0) class

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The practice problems in this post focus on counting distributions that belong to the (a,b,0) class, reinforcing the concepts discussed in this blog post in a companion blog.

The (a,b,1) class is a generalization of (a,b,0) class. It is discussed here. A practice problem set on the (a,b,1) class is found here.

Notation: $p_k=P(X=k)$ for $k=0,1,2,\cdots$ where $X$ is the counting distribution being focused on.

 Practice Problem 11-A Suppose that claim frequency $X$ follows a negative binomial distribution with parameters $r$ and $\theta$. The following is the probability function. $\displaystyle P(X=k)=\binom{r+k-1}{k} \ \biggl(\frac{1}{1+\theta} \biggr)^r \biggl(\frac{\theta}{1+\theta} \biggr)^k \ \ \ \ \ \ k=0,1,2,\cdots$ Evaluate the negative binomial distribution in two ways. Evaluate $P(X=k)$ for $k=0,1,2,3,4$ with $r=\frac{11}{6}$ and $\theta=1$. Convert $r=\frac{11}{6}$ and $\theta=1$ into the parameters $a$ and $b$. Evaluate the (a,b,0) distribution for $k=1,2,3,4$.
 Practice Problem 11-B Suppose that $X$ follows a distribution in the (a,b,0) class. You are given that $p_2=0.185351532$ $p_3=0.105032535$ $p_4=0.055142081$ Evaluate the probability that $X$ is at least 1.
 Practice Problem 11-C The following information is given about a distribution from the (a,b,0) class. $p_3=0.160670519$ $p_4=0.072301734$ $p_5=0.026028624$ What is the form of the distribution? Evaluate $p_1$.
 Practice Problem 11-D For a distribution from the (a,b,0) class, the following information is given. $p_1=0.214663$ $p_2=0.053666$ $p_3=0.012522$ Determine the variance of this distribution.
 Practice Problem 11-E For a distribution from the (a,b,0) class, you are given that $a=-1/3$ and $b=2$. Find the value of $p_0$.
 Practice Problem 11-F You are given that the distribution for the claim count $X$ satisfies the following recursive relation: $\displaystyle p_k=\frac{2 p_{k-1}}{k} \ \ \ \ \ \ \ \ \ k=1,2,3,\cdots$ Determine $P[X=2]$.
 Practice Problem 11-G Suppose that the random variable $X$ is from the (a,b,0) class. You are given that $a=-1/4$ and $b=7/4$. Calculate the probability that $X$ is at least 3.
 Practice Problem 11-H For a distribution from the (a,b,0) class, you are given that $p_2=0.20736$ $p_3=0.13824$ $p_4=0.082944$ Determine $p_1$.
 Practice Problem 11-I For a distribution from the (a,b,0) class, you are given that $a=1/6$ and $b=1/2$. Evaluate its mean.
 Practice Problem 11-J The random variable $X$ follows a distribution from the (a,b,0) class. You are given that $a=0.6$ and $b=-0.3$. Evaluate $E(X^2)$.
 Practice Problem 11-K The random variable $X$ follows a distribution from the (a,b,0) class. Suppose that $E(X)=3$ and $Var(X)=12$. Determine $p_2$.
 Practice Problem 11-L Given that a discrete distribution is a member of the (a,b,0) class. Which of the following statement(s) are true? If $a>0$ and $b>0$, then the variance of the distribution is greater than the mean. If $a>0$ and $b=0$, then the variance of the distribution is less than the mean. If $a<0$ and $b>0$, then the variance of the distribution is greater than the mean. A. ……….. 1 only B. ……….. 2 only C. ……….. 3 only D. ……….. 1 and 2 only E. ……….. 1 and 3 only
 Practice Problem 11-M Given that a discrete distribution is a member of the (a,b,0) class, determine the variance of the distribution if $a=1/6$ and $b=1/4$.
 Practice Problem 11-N For a distribution in the (a,b,0) class, $p_2=0.2048$ and $p_3=0.0512$. Furthermore, the mean of the distribution is 1. Determine $p_1$. $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

11-A
• $a=1/2$ and $b=5/12$
• $\displaystyle p_0=\biggl(\frac{1}{2} \biggr)^{11/6}$
• $\displaystyle p_1=\biggl(\frac{11}{12} \biggr) \biggl(\frac{1}{2} \biggr)^{11/6}$
• $\displaystyle p_2=\biggl(\frac{17}{24} \biggr) \biggl(\frac{11}{12} \biggr) \biggl(\frac{1}{2} \biggr)^{11/6}$
• $\displaystyle p_3=\biggl(\frac{23}{36} \biggr) \biggl(\frac{17}{24} \biggr) \biggl(\frac{11}{12} \biggr) \biggl(\frac{1}{2} \biggr)^{11/6}$
• $\displaystyle p_4=\biggl(\frac{29}{48} \biggr) \biggl(\frac{23}{36} \biggr) \biggl(\frac{17}{24} \biggr) \biggl(\frac{11}{12} \biggr) \biggl(\frac{1}{2} \biggr)^{11/6}$
11-B
• $1-(0.6)^{2.25}=0.683159775$
11-C
• Poisson distribution with mean 1.8
• $1.8 e^{-1.8}=0.2975$
11-D
• 0.46875
11-E
• $\displaystyle p_0=\frac{243}{1024}=0.237305$
11-F
• $2 e^{-2}=0.270671$
11-G
• 0.09888
11-H
• 0.2592
11-I
• 0.8
11-J
• 2.4375
11-K
• $\displaystyle p_2=\frac{5.625}{256}=0.02197$
11-L
• A. 1 only
11-M
• 0.6
11-N
• 0.4096

Daniel Ma Math

Daniel Ma Mathematics

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