随机变量 $X$ 服从分布 $\mathrm{N}(\mu, \sigma^2)$。

1. Identify the symmetrical relationship between $2\mu - k$ and $k$ about the mean $\mu$.

The normal distribution curve is symmetrical about $x = \mu$.
The distance from $2\mu - k$ to the mean $\mu$ is $\mu - (2\mu - k) = k - \mu$, which is exactly the same as the distance from $\mu$ to $k$. Thus, $2\mu - k$ is the reflection of $k$ across the mean.

2. Apply the symmetry property and the complement rule to calculate the probability.

By symmetry, the probability to the right of $2\mu - k$ is equal to the probability to the left of $k$:
$\mathrm{P}(X > 2\mu - k) = \mathrm{P}(X < k)$
Using the complement rule:
$\mathrm{P}(X < k) = 1 - \mathrm{P}(X > k)$
$1 - 0.45 = 0.55$

1. Find the probability that $X < 1$.

Since $\mu = 4$, the distribution is symmetric about $X = 4$, so $\mathrm{P}(X < 4) = 0.5$.
$\mathrm{P}(X < 1) = \mathrm{P}(X < 4) - \mathrm{P}(1 < X < 4)$
$\mathrm{P}(X < 1) = 0.5 - 0.2475 = 0.2525$

2. Standardise the value and solve for $\sigma$ using the inverse normal distribution.

Standardising using $Z = \frac{X - \mu}{\sigma}$:
$\mathrm{P}\left(Z < \frac{1 - 4}{\sigma}\right) = 0.2525$
Using the inverse normal function on a graphing calculator:
$\frac{-3}{\sigma} \approx -0.666665$
$\sigma = \frac{-3}{-0.666665} \approx 4.50$

1. Calculate the standard deviation $\sigma$ using information from part (b).

$\mathrm{P}(1 < X < 4) = 0.2475$
Since the normal curve is symmetrical about the mean $\mu = 4$, $\mathrm{P}(X < 4) = 0.5$.
$\mathrm{P}(X < 1) = 0.5 - 0.2475 = 0.2525$
Standardising the value:
$\mathrm{P}\left(Z < \frac{1 - 4}{\sigma}\right) = 0.2525$
$\frac{-3}{\sigma} = \Phi^{-1}(0.2525) \approx -0.66667$
$\sigma = 4.5$

2. Determine the features of the sketch relative to the mean and standard deviation.

The normal curve is symmetrical and peaks at the mean, $x = 4$.
The domain boundaries $-11$ and $19$ are equidistant from the mean:
$4 - (-11) = 15 \quad \text{and} \quad 19 - 4 = 15$
Since $15 = \frac{15}{4.5}\sigma \approx 3.33\sigma$, the required interval $[-11, 19]$ covers more than $3$ standard deviations on either side of the mean. Therefore, the curve should practically taper off to the $x$-axis at these boundaries.

3. Describe the required sketch.

• Draw a horizontal $x$-axis.
• Sketch a bell-shaped curve strictly above the $x$-axis.
• Mark the maximum point on the curve at the line of symmetry $x = 4$.
• Extend the tails so they taper off and closely approach the $x$-axis at the labeled endpoints $x = -11$ and $x = 19$.