* Sample Space ($\Omega$ or $S$): The set of all possible outcomes of a random experiment. For example, if you roll a die, the sample space is $\{1, 2, 3, 4, 5, 6\}$.

* Event: Any subset of the sample space. An event is a collection of one or more outcomes. For example, "rolling an even number" is an event $\{2, 4, 6\}$.

* Mutually Exclusive Events: Two events, A and B, are mutually exclusive (or disjoint) if they cannot occur at the same time. Their intersection is empty ($A \cap B = \emptyset$).

* For any two events A and B: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

* If A and B are mutually exclusive: $P(A \cup B) = P(A) + P(B)$

* For any two events A and B: $P(A \cap B) = P(A|B)P(B)$

* If A and B are independent: $P(A \cap B) = P(A)P(B)$

* $P(A|B) = \frac{P(A \cap B)}{P(B)}$, provided $P(B) > 0$.

* Properties: $0 \le P(x) \le 1$ for all $x$, and $\sum_{x} P(x) = 1$.

* Properties: $f(x) \ge 0$ for all $x$, and $\int_{-\infty}^{\infty} f(x) dx = 1$.

* For discrete RV: $F(x) = P(X \le x) = \sum_{t \le x} P(t)$

* For continuous RV: $F(x) = P(X \le x) = \int_{-\infty}^{x} f(t) dt$

* Expected Value ($E(X)$ or $\mu$): The long-run average value of a random variable.

* For discrete RV: $E(X) = \sum_{x} x P(x)$

* For continuous RV: $E(X) = \int_{-\infty}^{\infty} x f(x) dx$

* Variance ($Var(X)$ or $\sigma^2$): A measure of the spread or dispersion of the distribution around its mean.

* $Var(X) = E[(X - \mu)^2] = E(X^2) - (E(X))^2$

* For discrete RV: $Var(X) = \sum_{x} (x - \mu)^2 P(x)$

* For continuous RV: $Var(X) = \int_{-\infty}^{\infty} (x - \mu)^2 f(x) dx$

* Standard Deviation ($\sigma$): The square root of the variance, providing a measure of spread in the original units of the random variable. $\sigma = \sqrt{Var(X)}$.

* Parameters: $n$ (number of trials), $p$ (probability of success).

* PMF: $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$, for $k = 0, 1, \dots, n$.

* Mean: $E(X) = np$

* Variance: $Var(X) = np(1-p)$

* Parameter: $\lambda$ (average rate of events).

* PMF: $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$, for $k = 0, 1, 2, \dots$.

* Mean: $E(X) = \lambda$

* Variance: $Var(X) = \lambda$

* Parameters: $\mu$ (mean), $\sigma$ (standard deviation).

* PDF: $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}$

* Standard Normal Distribution: A special case with $\mu=0$ and $\sigma=1$. Any normal variable $X$ can be standardized using $Z = \frac{X - \mu}{\sigma}$.

* Parameter: $\lambda$ (rate parameter, inverse of mean time between events).

* PDF: $f(x) = \lambda e^{-\lambda x}$, for $x \ge 0$.

* CDF: $F(x) = 1 - e^{-\lambda x}$, for $x \ge 0$.

* Mean: $E(X) = 1/\lambda$

* Variance: $Var(X) = 1/\lambda^2$

* Parameters: $a$ (minimum value), $b$ (maximum value).

* PDF: $f(x) = \frac{1}{b-a}$, for $a \le x \le b$, and $0$ otherwise.

* Mean: $E(X) = \frac{a+b}{2}$

* Variance: $Var(X) = \frac{(b-a)^2}{12}$

* Mean of the sampling distribution of $\bar{X}$: $E(\bar{X}) = \mu$ (population mean).

* Standard deviation of the sampling distribution of $\bar{X}$ (Standard Error of the Mean): $\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}$ (where $\sigma$ is the population standard deviation).

* If the population is normal, the sampling distribution of $\bar{X}$ is normal for any sample size $n$.

* Confidence Interval for Population Mean ($\mu$) (Large Sample, $\sigma$ known):

* Formula: $\bar{x} \pm Z_{\alpha/2} \left(\frac{\sigma}{\sqrt{n}}\right)$

* Where: $\bar{x}$ is the sample mean, $\sigma$ is the population standard deviation, $n$ is the sample size, and $Z_{\alpha/2}$ is the critical Z-value for the desired confidence level (e.g., 1.96 for 95% CI).

* Confidence Interval for Population Mean ($\mu$) (Large Sample, $\sigma$ unknown):

* Formula: $\bar{x} \pm Z_{\alpha/2} \left(\frac{s}{\sqrt{n}}\right)$

* Where $s$ is the sample standard deviation. (Note: For smaller samples or when $\sigma$ is unknown, the t-distribution is typically used, but the questions imply large sample Z-intervals).

* Null Hypothesis ($H_0$): A statement of no effect or no difference, often representing the status quo.

* Alternative Hypothesis ($H_1$ or $H_a$): A statement that contradicts the null hypothesis, representing what the researcher wants to prove.

2. Set Significance Level ($\alpha$): The probability of rejecting the null hypothesis when it is actually true (Type I error). Common values are 0.05 or 0.01.

3. Choose Test Statistic: A value calculated from sample data that is used to decide whether to reject $H_0$. The choice depends on the parameter being tested, sample size, and knowledge of population variance.

* Z-test for Population Mean (Large Sample): $Z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$ (if $\sigma$ is known) or $Z = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$ (if $\sigma$ is unknown but $n$ is large).

* Critical Region: The range of values for the test statistic that would lead to rejection of $H_0$.

* P-value: The probability of observing a test statistic as extreme as, or more extreme than, the one calculated from the sample, assuming $H_0$ is true.

* If the test statistic falls in the critical region, or if P-value $\le \alpha$, reject $H_0$.

* Otherwise, fail to reject $H_0$.

* Type I Error ($\alpha$): Rejecting a true null hypothesis. This is often called a "false positive".

* Type II Error ($\beta$): Failing to reject a false null hypothesis. This is often called a "false negative".

* Range: $-1 \le r \le 1$.

* $r=1$: Perfect positive linear relationship.

* $r=-1$: Perfect negative linear relationship.

* $r=0$: No linear relationship.

* Formula (Pearson product-moment correlation coefficient): $r = \frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}$

* Regression Equation: $\hat{y} = a + bx$

* $\hat{y}$: The predicted value of the dependent variable.

* $a$: The y-intercept, the predicted value of $Y$ when $X=0$.

* $b$: The slope, the predicted change in $Y$ for a one-unit increase in $X$.

* Formulas for Slope ($b$) and Y-intercept ($a$):

* $b = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}$

* $a = \bar{y} - b\bar{x}$

* Coefficient of Determination ($R^2$): The proportion of the variance in the dependent variable ($Y$) that is predictable from the independent variable ($X$). It is the square of the correlation coefficient ($R^2 = r^2$).

* Range: $0 \le R^2 \le 1$. A higher $R^2$ indicates a better fit of the model to the data.

* Expected Value ($E(X)$): The long-run average value of a random variable, representing the center of its distribution. It's a weighted average of all possible values.

* Variance ($Var(X)$): A measure of the spread or dispersion of a random variable's distribution around its expected value. A higher variance indicates greater variability.

* Standard Deviation ($\sigma$): The square root of the variance, providing a measure of spread in the same units as the random variable itself, making it more interpretable than variance.

* Null Hypothesis ($H_0$): A statement in hypothesis testing that assumes no effect, no difference, or no relationship between variables. It is the hypothesis that researchers aim to test against.

* Alternative Hypothesis ($H_1$ or $H_a$): A statement that contradicts the null hypothesis, representing the effect, difference, or relationship that the researcher is trying to find evidence for.

* Significance Level ($\alpha$): The probability of making a Type I error, i.e., rejecting the null hypothesis when it is actually true. It is a threshold set by the researcher.

* P-value: The probability of observing a test statistic as extreme as, or more extreme than, the one calculated from the sample, assuming the null hypothesis is true. A small p-value suggests evidence against $H_0$.

* Type I Error: The error of rejecting a true null hypothesis (false positive). Its probability is denoted by $\alpha$.

* Type II Error: The error of failing to reject a false null hypothesis (false negative). Its probability is denoted by $\beta$.

* Correlation Coefficient ($r$): A statistical measure that quantifies the strength and direction of a linear relationship between two quantitative variables.

* Coefficient of Determination ($R^2$): The proportion of the variance in the dependent variable that can be explained by the independent variable(s) in a regression model. It indicates how well the model fits the data.

* Explanation: This is a binomial distribution problem with $n=10$ (trials) and $p=0.05$ (probability of success/defect). We use the PMF: $P(X=2) = \binom{10}{2} (0.05)^2 (0.95)^{10-2}$.

* Explanation: This is a Poisson distribution problem with $\lambda=3$ (average rate). We use the PMF: $P(X=5) = \frac{e^{-3} 3^5}{5!}$.

* Explanation: We standardize the values using the Z-score formula: $Z_1 = (60-75)/10 = -1.5$ and $Z_2 = (80-75)/10 = 0.5$. Then, we find $P(-1.5 \le Z \le 0.5)$ using a standard normal table or calculator.

* Explanation: The mean is $1/\lambda = 500$, so $\lambda = 1/500 = 0.002$. We use the CDF: $P(X \le 200) = 1 - e^{-(0.002)(200)} = 1 - e^{-0.4}$.

* Explanation: Since $n=100$ (large sample) and $s=8$ (sample standard deviation used as an estimate for $\sigma$), we use the Z-interval formula: $\bar{x} \pm Z_{\alpha/2} (s/\sqrt{n})$. For 95% confidence, $Z_{\alpha/2} = 1.96$. So, $165 \pm 1.96 (8/\sqrt{100}) = 165 \pm 1.96(0.8)$.

1. $H_0: \mu = 1000$, $H_1: \mu \ne 1000$.

2. $\alpha = 0.05$.

3. Test statistic (Z-test for mean, large sample): $Z = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} = \frac{980 - 1000}{75/\sqrt{50}}$.

4. Calculate Z-value and compare to critical Z-values ($\pm 1.96$ for two-tailed 5% test) or calculate p-value.

5. Make a decision (reject or fail to reject $H_0$) and conclude whether there's enough evidence to dispute the company's claim.

* Example: Given data on advertising expenditure ($X$) and sales ($Y$), calculate the regression equation $\hat{y} = a + bx$ and predict sales for a given advertising expenditure.

* Explanation: Use the formulas for $b$ and $a$ based on sums of $X$, $Y$, $X^2$, $Y^2$, and $XY$. For instance, if the calculated equation is $\hat{y} = 10 + 2.5x$, and advertising expenditure is 10 units, predicted sales would be $\hat{y} = 10 + 2.5(10) = 35$ units. Also, interpret the slope (e.g., for every unit increase in advertising, sales are predicted to increase by 2.5 units).

The document begins with Basic Probability Theory, emphasizing definitions such as sample space, events, mutually exclusive events, and independent events. It reinforces the core rules of probability, including the addition rule for unions of events ($P(A \cup B) = P(A) + P(B) - P(A \cap B)$) and the multiplication rule for intersections ($P(A \cap B) = P(A|B)P(B)$), with special consideration for mutually exclusive and independent cases. Conditional probability ($P(A|B) = P(A \cap B)/P(B)$) is also a key concept.

Following this, the document delves into Random Variables and Probability Distributions. It differentiates between discrete and continuous random variables and introduces their respective probability functions: the Probability Mass Function (PMF) for discrete variables (where $\sum P(x) = 1$) and the Probability Density Function (PDF) for continuous variables (where $\int f(x) dx = 1$). The Cumulative Distribution Function (CDF), $F(x) = P(X \le x)$, is presented as a unifying concept for both types. Crucially, the concepts of Expected Value ($E(X)$ or mean) and Variance ($Var(X)$ or $\sigma^2$) are explained as measures of central tendency and dispersion, respectively, with their corresponding formulas for both discrete and continuous cases. The standard deviation ($\sigma = \sqrt{Var(X)}$) is also highlighted for its interpretability.

Specific probability distributions are thoroughly covered. For discrete distributions, the Binomial distribution (for fixed trials, $P(X=k)