BSc CSIT (TU) Science Statistics II (BSc CSIT, STA210) Question Paper 2079 Nepal

Probability Distribution

A probability distribution is a mathematical function (or table) that assigns to each possible value of a random variable the probability of its occurrence. For a discrete random variable $X$ taking values $x_1, x_2, \dots$ it is described by the probability mass function $P(X = x_i) = p_i$ with $p_i \ge 0$ and $\sum_i p_i = 1$; for a continuous variable it is described by a probability density function $f(x)$ with $f(x) \ge 0$ and $\int_{-\infty}^{\infty} f(x)\,dx = 1$.

Binomial Distribution

A discrete random variable $X$ follows a binomial distribution if it counts the number of successes in $n$ independent Bernoulli trials, each having success probability $p$ (and failure $q = 1 - p$). Its probability mass function is

Mean and Variance

Since $q < 1$, the variance $npq$ is always less than the mean $np$, i.e. for the binomial distribution mean $>$ variance.

Outline of the mean: writing $X = \sum_{i=1}^{n} X_i$ where each $X_i$ is Bernoulli with $E(X_i)=p$, by linearity $E(X)=\sum E(X_i)=np$; similarly $\operatorname{Var}(X_i)=pq$ and by independence $\operatorname{Var}(X)=npq$.

Conditions for Application

The binomial distribution applies when:

1. The number of trials $n$ is fixed and finite.
2. Each trial has only two outcomes — success or failure (dichotomous).
3. The trials are independent of one another.
4. The probability of success $p$ remains constant from trial to trial.

Examples: number of heads in 10 tosses of a coin, number of defective items in a sample of fixed size, number of correct answers in a multiple-choice test.

Normal Distribution

A continuous random variable $X$ follows a normal distribution with mean $\mu$ and standard deviation $\sigma$ if its probability density function is

We write $X \sim N(\mu, \sigma^2)$.

Properties

1. The curve is bell-shaped and symmetric about the mean $\mu$.
2. Mean = Median = Mode = $\mu$.
3. It is unimodal; the maximum of $f(x)$ occurs at $x = \mu$.
4. The curve is asymptotic to the $x$-axis on both sides.
5. Total area under the curve is 1, with half on each side of $\mu$.
6. The points of inflection occur at $x = \mu \pm \sigma$.
7. Empirical rule: about 68.27%, 95.45% and 99.73% of values lie within $\mu\pm\sigma$, $\mu\pm 2\sigma$ and $\mu\pm 3\sigma$ respectively.
8. Quartile deviation $= 0.6745\sigma$ and mean deviation $= 0.7979\sigma$.
9. The standard normal variate is $Z = \dfrac{X - \mu}{\sigma} \sim N(0,1)$.

Numerical: $P(45 < X < 62)$

Given $\mu = 50$, $\sigma = 10$. Convert to $Z$-scores:

Using the standard normal table ($\Phi$ = area from $0$ to $z$):

Since the two points lie on opposite sides of the mean,

Hence the required probability is about 0.5764 (57.64%).

Hypothesis Testing

Hypothesis testing is a statistical procedure used to decide, on the basis of sample data, whether to accept or reject an assumption (hypothesis) made about a population parameter. It quantifies how strongly the sample evidence contradicts the assumption.

Key Concepts

1. Null Hypothesis ($H_0$): A statement of no effect or no difference that is assumed true until evidence suggests otherwise, e.g. $H_0: \mu = \mu_0$.

2. Alternative Hypothesis ($H_1$): The statement accepted if $H_0$ is rejected. It may be:

• Two-tailed: $H_1: \mu \ne \mu_0$
• Right-tailed: $H_1: \mu > \mu_0$
• Left-tailed: $H_1: \mu < \mu_0$

3. Level of Significance ($\alpha$): The maximum probability of rejecting $H_0$ when it is actually true (probability of a Type I error). Common values are $0.05$ (5%) and $0.01$ (1%).

4. Types of Errors:

• Type I error ($\alpha$): rejecting a true $H_0$.
• Type II error ($\beta$): accepting a false $H_0$. The quantity $1 - \beta$ is the power of the test.

5. Critical Region (Rejection Region): The set of values of the test statistic for which $H_0$ is rejected. Its area equals $\alpha$. The boundary value(s) separating it from the acceptance region are the critical value(s).

Procedure (Steps)

1. Set up the null hypothesis $H_0$ and alternative hypothesis $H_1$.
2. Choose the level of significance $\alpha$.
3. Select an appropriate test statistic (Z, t, $\chi^2$, F) based on sample size and what is known.
4. Determine the critical value and the critical (rejection) region from statistical tables.
5. Compute the value of the test statistic from the sample data.
6. Decision: If the computed value falls in the critical region, reject $H_0$; otherwise accept (fail to reject) $H_0$.
7. State the conclusion in the context of the problem.

Addition Theorem of Probability

For any two events $A$ and $B$,

If $A$ and $B$ are mutually exclusive ($A \cap B = \varnothing$), this reduces to

Example: Drawing one card from a pack, $P(\text{King or Queen}) = \frac{4}{52} + \frac{4}{52} = \frac{8}{52} = \frac{2}{13}$ (mutually exclusive). For $P(\text{King or Heart}) = \frac{4}{52} + \frac{13}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}$ (not mutually exclusive).

Multiplication Theorem of Probability

For any two events $A$ and $B$,

where $P(B \mid A)$ is the conditional probability of $B$ given $A$. If $A$ and $B$ are independent, then $P(B\mid A) = P(B)$ and

Example: Two cards drawn one after another without replacement, $P(\text{both Kings}) = \frac{4}{52} \times \frac{3}{51} = \frac{1}{221}$. Tossing two coins (independent), $P(\text{both heads}) = \frac{1}{2}\times\frac{1}{2} = \frac{1}{4}$.

Poisson Distribution

The Poisson distribution is a discrete distribution that models the number of times a rare event occurs in a fixed interval of time, space or volume when occurrences are independent and at a constant average rate. A random variable $X$ follows a Poisson distribution with parameter $\lambda$ (the mean number of occurrences) if

It is the limiting form of the binomial distribution when $n \to \infty$, $p \to 0$, with $np = \lambda$ finite.

Mean and Variance

A characteristic feature is that the mean equals the variance $(\mu = \sigma^2 = \lambda)$.

Applications

Used for counting rare, independent events such as:

1. Number of telephone calls arriving at an exchange per minute.
2. Number of printing/typing errors per page of a book.
3. Number of accidents on a highway per day.
4. Number of defective items in a large batch.
5. Number of customers (or network packets/requests) arriving at a server per unit time.
6. Number of radioactive particles emitted per second.

Random Variable

A random variable is a real-valued function that assigns a numerical value to each outcome (sample point) of a random experiment. It is usually denoted by a capital letter $X$, and a particular value by a small letter $x$. For example, if a coin is tossed twice and $X$ denotes the number of heads, then $X$ takes the values $0, 1, 2$.

Discrete vs Continuous Random Variables

Examples

• Discrete: number of students present in a class, number of cars passing a point in an hour.
• Continuous: the exact weight of a person, the lifetime of an electric bulb, daily rainfall.

Mathematical Expectation

The mathematical expectation (or expected value) of a random variable is the long-run average value it takes, weighted by probabilities. For a discrete random variable $X$ with pmf $p(x)$,

and for a continuous random variable with pdf $f(x)$,

Properties (with proofs)

1. Expectation of a constant. If $k$ is a constant, $E(k) = k$.

Proof: $E(k) = \sum_x k\,p(x) = k\sum_x p(x) = k\cdot 1 = k.$

2. Multiplication by a constant. $E(kX) = k\,E(X)$.

Proof: $E(kX) = \sum_x kx\,p(x) = k\sum_x x\,p(x) = k\,E(X).$

3. Addition (linearity). $E(X + Y) = E(X) + E(Y)$ for any random variables.

Proof (discrete): $E(X+Y) = \sum_x\sum_y (x+y)p(x,y) = \sum_x\sum_y x\,p(x,y) + \sum_x\sum_y y\,p(x,y) = E(X) + E(Y).$

Combining 2 and 3: $E(aX + b) = aE(X) + b$.

4. Product for independent variables. If $X$ and $Y$ are independent, $E(XY) = E(X)\,E(Y)$.

Proof: For independent variables $p(x,y) = p(x)p(y)$, so $E(XY) = \sum_x\sum_y xy\,p(x)p(y) = \left(\sum_x x\,p(x)\right)\left(\sum_y y\,p(y)\right) = E(X)E(Y).$

t-test for Difference of Two Sample Means

This test checks whether two independent small samples (sizes $n_1, n_2 < 30$) drawn from normal populations with equal (unknown) variances have significantly different means.

Hypotheses: $H_0: \mu_1 = \mu_2$ (no difference) against $H_1: \mu_1 \ne \mu_2$ (two-tailed) or one-sided alternative.

Test statistic:

where the pooled standard deviation is

The statistic follows the t-distribution with $\nu = n_1 + n_2 - 2$ degrees of freedom.

Decision rule: Compute $|t|$ and compare with the table value $t_{\alpha,\nu}$ at significance level $\alpha$.

• If $|t| \le t_{\alpha,\nu}$: accept $H_0$ — the difference is not significant.
• If $|t| > t_{\alpha,\nu}$: reject $H_0$ — the difference is significant.

Assumptions: the parent populations are normal, the two samples are independent, and the population variances are equal (homogeneous).

z-test for a Single Mean (Large Sample)

When the sample size is large ($n \ge 30$), the sampling distribution of the mean is approximately normal, so a z-test is used to test whether a sample mean $\bar{x}$ differs significantly from a hypothesised population mean $\mu$.

Hypotheses: $H_0: \mu = \mu_0$ versus $H_1: \mu \ne \mu_0$ (two-tailed).

Test statistic:

where $\sigma$ is the population standard deviation (if unknown, the sample s.d. $s$ is used since $n$ is large). $Z$ follows the standard normal distribution.

Decision rule (5% level): reject $H_0$ if $|Z| > 1.96$ (for 1%, if $|Z| > 2.58$); otherwise accept $H_0$.

Example

A sample of $n = 64$ items has mean $\bar{x} = 52$, drawn from a population with mean $\mu_0 = 50$ and $\sigma = 8$. Test at 5% whether the sample mean differs from 50.

Since $|Z| = 2.0 > 1.96$, we reject $H_0$: the sample mean differs significantly from 50 at the 5% level of significance.

Karl Pearson's Coefficient of Correlation

Correlation measures the degree and direction of the linear relationship between two variables $X$ and $Y$. Karl Pearson's coefficient of correlation (product-moment correlation), denoted $r$, is defined as

An equivalent computational form is

Properties

1. Range: $r$ always lies between $-1$ and $+1$, i.e. $-1 \le r \le +1$.
2. Interpretation: $r = +1$ perfect positive, $r = -1$ perfect negative, $r = 0$ no linear correlation.
3. Independent of origin and scale (units): $r$ is unchanged if each value is shifted or multiplied by a constant; it is a pure number.
4. Symmetric: $r_{XY} = r_{YX}$.
5. It is the geometric mean of the two regression coefficients: $r = \pm\sqrt{b_{xy}\,b_{yx}}$, taking the sign of the regression coefficients.
6. If $X$ and $Y$ are independent then $r = 0$ (the converse is not necessarily true).

Regression Coefficients

In linear regression between two variables $X$ and $Y$, the regression coefficient is the slope of the line of regression — it measures the average change in the dependent variable per unit change in the independent variable.

• Regression coefficient of $Y$ on $X$:

• Regression coefficient of $X$ on $Y$:

Properties

1. Correlation as geometric mean: the correlation coefficient is the geometric mean of the two regression coefficients,

2. Same sign: both regression coefficients have the same sign, which is also the sign of $r$. If one is positive both are positive, and so is $r$.
3. Product $\le 1$: since $-1 \le r \le 1$, $\;b_{xy}\cdot b_{yx} = r^2 \le 1$. Hence both coefficients cannot exceed unity simultaneously; if one is greater than 1, the other must be less than 1.
4. Independent of change of origin but not of scale.
5. The arithmetic mean of the two regression coefficients is greater than or equal to $r$, i.e. $\dfrac{b_{xy}+b_{yx}}{2} \ge r$.
6. The two regression lines intersect at the point $(\bar{x}, \bar{y})$.

Sampling Distribution

When all possible samples of a fixed size $n$ are drawn from a population, a statistic (such as the sample mean $\bar{x}$, proportion $p$, or variance $s^2$) varies from sample to sample. The probability distribution of all possible values of that statistic is called its sampling distribution.

For example, the sampling distribution of the mean lists every possible $\bar{x}$ with its probability. By the Central Limit Theorem, for large $n$ the sampling distribution of $\bar{x}$ is approximately normal with mean $\mu$ and standard deviation $\sigma/\sqrt{n}$, regardless of the shape of the parent population.

Standard Error (S.E.)

The standard error of a statistic is the standard deviation of its sampling distribution. It measures the variability of the statistic due to sampling and indicates how precisely the sample statistic estimates the population parameter.

Common standard errors:

• Mean: $\;\text{S.E.}(\bar{x}) = \dfrac{\sigma}{\sqrt{n}}$
• Proportion: $\;\text{S.E.}(p) = \sqrt{\dfrac{PQ}{n}}$, where $Q = 1-P$.

Uses: the standard error is used (i) to test the significance of a statistic (test statistic = (estimate − parameter)/S.E.), and (ii) to construct confidence intervals. A smaller standard error (larger $n$) means a more reliable estimate.