Formulario Esame Impianti Industriali

It appears you've provided a detailed set of notes on statistical methods and market forecasting techniques, particularly focusing on the binomial distribution, normal distribution, and various prediction methods. Here's a summary and explanation of the key points: ### Binomial Distribution - **Definition**: The binomial distribution is used to model the number of successes in a fixed number of independent Bernoulli trials (each with two possible outcomes: success or failure). - **Parameters**: - \( n \): Number of trials. - \( p \): Probability of success in each trial. - **Probability Mass Function**: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] - **Mean and Variance**: - Mean: \( E[X] = np \) - Variance: \( Var(X) = np(1-p) \) ### Normal Distribution - **Properties**: The normal distribution is a continuous probability distribution that is symmetric around the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. - **Standard Deviation and Z-Score**: - \( z = \frac{x - \mu}{\sigma} \) - **Confidence Interval for Proportions**: \[ \hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \] where \( \hat{p} \) is the sample proportion, and \( n \) is the sample size. ### Market Forecasting Techniques #### Correlation Method - **Formula for Slope (b)**: \[ b = \frac{\sum [(\text{X}_i - \bar{\text{X}})(\text{Y}_i - \bar{\text{Y}})]}{\sum (\text{X}_i - \bar{\text{X}})^2} \] - **Formula for Intercept (a)**: \[ a = \bar{\text{Y}} - b \cdot \bar{\text{X}} \] - **Prediction Formula**: \[ \hat{\text{Y}}_i = a + b(\text{X}_i) \] #### Method of Last Period (MUP) - **Formula**: \[ \text{Forecast}_{t+1} = \text{Demand}_t \] #### Simple Moving Average (SMA) Method - **Formula**: \[ \hat{\text{Y}}_{t+1} = \frac{\sum_{i=1}^{n} \text{Y}_{t-i+1}}{n} \] ### Example Application Given the example of promoting a new agricultural machine to 1750 farmers out of a total population of 15,000: - **Sample Proportion** (\( \hat{p} \)): \[ \hat{p} = \frac{1075}{1750} = 0.6143 \] - **Standard Error (SE)**: \[ SE = 2 \sqrt{\frac{0.6143 \cdot 0.3857}{1750