For emergency room receives on average 8 patients per hour. Calculate the mean and variance of Binomial and Poisson distribution.

The variance of a Poisson distribution is also equal to its rate parameter $\lambda$.
$$
Var(X) = \lambda
$$
$$
Var(X) = 8
$$
The mean is $\boxed{8}$ and the variance is $\boxed{8}$ for the Poisson distribution.

• Binomial Distribution: To calculate the mean and variance for a Binomial distribution, we need two parameters: the number of trials ($n$) and the probability of success ($p$). This information is not provided in the problem statement. The scenario described (average rate of events over a period) is characteristic of a Poisson process, not a Binomial one, unless further assumptions are made about fixed trials and success probabilities. Therefore, we cannot calculate the mean and variance for a Binomial distribution with the given information.

2. Use the example above to complete a line function $g(x) = 2x + 1$.

The function $g(x) = 2x + 1$ is already a complete linear function. It represents a straight line with a slope of 2 and a y-intercept of 1. Without further instructions (e.g., to evaluate it at a specific point, graph it, or relate it to the previous problem in a defined way), there is nothing to "complete".

3. The mean serum uric acid level in a population is $5.0 \text{ mg/100ml}$ with a SD of 0.5. Define the normal range (mean $\pm 2$ SD).

Mean ($\mu$) $= 5.0 \text{ mg/100ml}$
Standard Deviation ($\sigma$) $= 0.5 \text{ mg/100ml}$

The normal range is defined as $\mu \pm 2\sigma$.

Lower bound $= 5.0 - 1.0 = 4.0$
Upper bound $= 5.0 + 1.0 = 6.0$
The normal range is $\boxed{[4.0, 6.0] \text{ mg/100ml}}$.

4. A patient has a systolic blood pressure (SBP) of $135 \text{ mmHg}$. If the population mean is $120 \text{ mmHg}$ and the standard deviation is $10 \text{ mmHg}$, find the Z.

Patient's SBP ($x$) $= 135 \text{ mmHg}$
Population mean ($\mu$) $= 120 \text{ mmHg}$
Standard deviation ($\sigma$) $= 10 \text{ mmHg}$

The Z-score is calculated as:

The Z-score is $\boxed{1.5}$.

5. The length of stay $X$ (in days) for a specific routine procedure in a clinic is uniformly distributed between 1 and 5 days. Calculate the expected length of stay.

For a uniform distribution $U(a, b)$, the lower bound is $a=1$ day and the upper bound is $b=5$ days.

The expected value $E(X)$ for a uniform distribution is given by:

The expected length of stay is $\boxed{3 \text{ days}}$.

6. A hospital determines that the length of stay (in days) for a specific procedure follows PDF $f(x) = \frac{1}{3}$ for $0 \le x \le 3$. Calculate the SD of length of stay.

The given PDF $f(x) = \frac{1}{3}$ for $0 \le x \le 3$ indicates a uniform distribution $U(a, b)$ with $a=0$ and $b=3$.

The variance $Var(X)$ for a uniform distribution is given by:

The standard deviation is the square root of the variance.

The standard deviation of the length of stay is $\boxed{\sqrt{0.75} \approx 0.866 \text{ days}}$.