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

Here are the solutions to the problems. 1. For emergency room receives on average 8 patients per hour. Calculate the mean and variance of Binomial and Poisson distribution. Poisson Distribution: The number of patients arriving in an emergency room per hour is typically modeled by a Poisson distribution, where the average rate is given. Given average rate = 8 patients per hour. Step 1: Calculate the mean of the Poisson distribution. The mean of a Poisson distribution is equal to its rate parameter . E(X) = E(X) = 8 Step 2: Calculate the variance of the Poisson distribution. The variance of a Poisson distribution is also equal to its rate parameter . Var(X) = Var(X) = 8 The mean is 8 and the variance is 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 mg/100ml with a SD of 0.5. Define the normal range (mean ± 2 SD). Step 1: Identify the given values. Mean () = 5.0 mg/100ml Standard Deviation () = 0.5 mg/100ml Step 2: Apply the formula for the normal range. The normal range is defined as ± 2. Normal Range = ± 2 Normal Range = 5.0 ± 2(0.5) Normal Range = 5.0 ± 1.0 Step 3: Calculate the lower and upper bounds of the range. Lower bound = 5.0 - 1.0 = 4.0 Upper bound = 5.0 + 1.0 = 6.0 The normal range is [4.0, 6.0] mg/100ml. 4. A patient has a systolic blood pressure (SBP) of 135 mmHg. If the population mean is 120 mmHg and the standard deviation is 10 mmHg, find the Z. Step 1: Identify the given values. Patient's SBP (x) = 135 mmHg Population mean () = 120 mmHg Standard deviation () = 10 mmHg Step 2: Use the Z-score formula. The Z-score is calculated as: Z = (x - )/() Z = (135 - 120)/(10) Z = (15)/(10) Z = 1.5 The Z-score is 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. Step 1: Identify the parameters of the uniform distribution. For a uniform distribution U(a, b), the lower bound is a=1 day and the upper bound is b=5 days. Step 2: Use the formula for the expected value of a uniform distribution. The expected value E(X) for a uniform distribution is given by: E(X) = (a+b)/(2) E(X) = (1+5)/(2) E(X) = (6)/(2) E(X) = 3 The expected length of stay is 3 days. 6. A hospital determines that the length of stay (in days) for a specific procedure follows PDF f(x) = (1)/(3) for 0 x 3. Calculate the SD of length of stay. Step 1: Identify the parameters of the uniform distribution from the PDF. The given PDF f(x) = (1)/(3) for 0 x 3 indicates a uniform distribution U(a, b) with a=0 and b=3. Step 2: Calculate the variance of the uniform distribution. The variance Var(X) for a uniform distribution is given by: Var(X) = ((b-a)^2)/(12) Var(X) = ((3-0)^2)/(12) Var(X) = (3^2)/(12) Var(X) = (9)/(12) Var(X) = (3)/(4) = 0.75 Step 3: Calculate the standard deviation (SD). The standard deviation is the square root of the variance. SD(X) = sqrt(Var(X)) SD(X) = sqrt(0.75) SD(X) ≈ 0.866 The standard deviation of the length of stay is sqrt(0.75) ≈ 0.866 days. 3 done, 2 left today. You're making progress.