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3. An insurance company offers its policyholders a number of different premium payment options. For a mandomly selected poliegholder, let \( X= \) the number of months betseen successive payments. The cdf of X is as follows THE DEPARTMENTS OF MATHEMATICS AND STATISTICS 2023/2024 HARMATTAN SEMESTER HSTA 206: Probability Distribution II \[ F(x)=\left\{\begin{array}{ll} 0 & x<1 \\ .30 & 1 \leq x<3 \\ .40 & 3 \leq x<4 \\ .45 & 4 \leq x<6 \\ .60 & 6 \leq x<12 \\ 1 & 12 \leq x \end{array}\right. \] 2. What is the pmf of X ? b. Using just the cdf, compute \( P_{r}(3 \leq X \leq 6) \) and \( P_{r}(4 \leq X) \). 4. Benford's Iaw arises in a variety of situations as a model for the first digit of a number: \[ P(x)=P(1 \text { st digit is } x)=\log _{10}\left(\frac{x+1}{x}\right), \quad x=1,2, \cdots, 9 \] a. Obtain the cdf of X , a nv following Benford's law. b. Using the odf, what is the probability that the leading digit is at most 3 ? At least 5 ?

          3. An insurance company offers its policyholders a number of different premium payment options. For a mandomly selected poliegholder, let \( X= \) the number of months betseen successive payments. The cdf of X is as follows
THE DEPARTMENTS OF MATHEMATICS AND STATISTICS 2023/2024 HARMATTAN SEMESTER
HSTA 206: Probability Distribution II
\[
F(x)=\left\{\begin{array}{ll}
0 & x<1 \\
.30 & 1 \leq x<3 \\
.40 & 3 \leq x<4 \\
.45 & 4 \leq x<6 \\
.60 & 6 \leq x<12 \\
1 & 12 \leq x
\end{array}\right.
\]
2. What is the pmf of X ?
b. Using just the cdf, compute \( P_{r}(3 \leq X \leq 6) \) and \( P_{r}(4 \leq X) \).
4. Benford's Iaw arises in a variety of situations as a model for the first digit of a number:
\[
P(x)=P(1 \text { st digit is } x)=\log _{10}\left(\frac{x+1}{x}\right), \quad x=1,2, \cdots, 9
\]
a. Obtain the cdf of X , a nv following Benford's law.
b. Using the odf, what is the probability that the leading digit is at most 3 ? At least 5 ?

3. An insurance company offers its policyholders a number of different premium payment options. For a mandomly selected poliegholder, let X= the number of months betseen successive payments. The cdf of X is as follows
THE DEPARTMENTS OF MATHEMATICS AND STATISTICS 2023/2024 HARMATTAN SEMESTER
HSTA 206: Probability Distribution II

F(x)={
0 x<1

.30 1 ≤ x<3

.40 3 ≤ x<4

.45 4 ≤ x<6

.60 6 ≤ x<12

1 12 ≤ x
.

2. What is the pmf of X ?
b. Using just the cdf, compute Pr(3 ≤ X ≤ 6) and Pr(4 ≤ X).
4. Benford's Iaw arises in a variety of situations as a model for the first digit of a number:

P(x)=P(1 st digit is x)=log10((x+1)/(x)), x=1,2, ⋯, 9

a. Obtain the cdf of X , a nv following Benford's law.
b. Using the odf, what is the probability that the leading digit is at most 3 ? At least 5 ?

Added by Kathy M.

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