= Solution
Record the discount level at the start of each policy year as the state. A <no claims discount system> is a <discrete-time Markov chain> when, conditional on this state, the next year's claim behavior and update rule do not depend on earlier history. Independent annual accident and loss mechanisms with a fixed state-based reporting rule give this property; one should not equate the probability of an accident with the probability of a submitted claim.
The density identifies $Z=(\log L-\mu)/\sigma$ as a <standard normal random variable>: substitute $x=e^{\mu+\sigma z}$ and $dx=\sigma x\,dz$. Therefore, for $a>0$,
$$
\boxed{\mathbb P(L>a)=1-\Phi\left(\frac{\log a-\mu}{\sigma}\right).}
$$
Here $L$ is the numerical repair cost. The character preceding $L$ in the PDF is a currency symbol, not a separate multiplier.
Order states as $0\%,20\%,30\%$. With the stipulated common annual probability $p$ of no submitted claims, the <transition matrix> is
$$
\boxed{P=\begin{pmatrix}
1-p&p&0\\
1-p&0&p\\
1-p&0&p
\end{pmatrix}.}
$$
A claim resets any state to zero, while a no-claim year raises the discount one level or keeps the top level fixed. If claim probabilities depend on the state, the corresponding row instead uses $p_i=\mathbb P(\text{no submitted claim}\mid\text{state }i)$; the same transition mechanism applies. The supplied single $p$ is a simplifying model assumption.
For a policyholder currently at zero discount, the current premium $C$ has the same value under either decision and cancels in the comparison. If the repair is self-funded and no later accidents occur, the next two premiums are $0.8C$ and $0.7C$, giving a total cost $L+2.5C$ including the current premium. If a claim is made, the next two premiums are $C$ and $0.8C$, giving total cost $2.8C$, with the repair paid by the insurer. Thus the <two-year no-claims threshold with reset to zero> is
$$
\boxed{L>0.3C.}
$$
At equality the holder is indifferent; it has probability zero under the continuous density. This assumes full reimbursement and no excess or deductible, as in the stated model.
Put $a=0.3C$. The conditional expected claim is $\mathbb E[L\mid L>a]$. Completing the square in the normal integral gives the <truncated lognormal moment>
$$
\mathbb E[L\mathbf1_{\{L>a\}}]
=\int_{z_a}^{\infty}e^{\mu+\sigma z}\frac{e^{-z^2/2}}{\sqrt{2\pi}}\,dz
=e^{\mu+\sigma^2/2}\left[1-\Phi(z_a-\sigma)\right],\qquad z_a=\frac{\log a-\mu}{\sigma}.
$$
Dividing by the reporting probability gives \b[the expected size of a submitted claim for the full-premium holder]:
$$
\boxed{\mathbb E[L\mid\text{claim}]=e^{\mu+\sigma^2/2}
\frac{1-\Phi\left((\log(0.3C)-\mu-\sigma^2)/\sigma\right)}
{1-\Phi\left((\log(0.3C)-\mu)/\sigma\right)}.}
$$
All monetary amounts must use the same unit despite the PDF's mixed currency notation. The other two current levels both have next-two-year no-claim premiums $0.7C+0.7C$, versus $C+0.8C$ after a claim, so their thresholds are $0.4C$. Consequently incorporating this behavioral decision into claim probabilities generally produces state-specific $p_i$, rather than deriving a common $p$ from the severity density alone; no accident-frequency law is provided.
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