BSc CSIT (TU) Science Statistics II (BSc CSIT, STA210) Question Paper 2074 Nepal

Probability Distribution

A probability distribution of a random variable $X$ is a description that assigns to every possible value (or interval of values) of $X$ a probability of occurrence, such that the total probability is $1$.

• Discrete case: a probability mass function $P(X=x_i)=p_i$ with $p_i\ge 0$ and $\sum_i p_i = 1$.
• Continuous case: a probability density function $f(x)$ with $f(x)\ge 0$ and $\int_{-\infty}^{\infty} f(x)\,dx = 1$.

Binomial Distribution

The binomial distribution describes the number of successes $X$ in a fixed number $n$ of independent trials, where each trial has only two outcomes (success/failure) with constant success probability $p$ (and $q = 1-p$).

Its probability mass function is:

Mean and Variance

Since $q<1$, the variance is always less than the mean for a binomial distribution.

Conditions for Application

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

Example: Tossing a fair coin $10$ times and counting heads follows $B(n=10,\,p=0.5)$.

Normal Distribution

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

Properties

1. The curve is bell-shaped and symmetric about the mean $\mu$.
2. Mean = Median = Mode $= \mu$.
3. Total area under the curve equals $1$; the curve is asymptotic to the $x$-axis.
4. It is defined by two parameters: $\mu$ (location) and $\sigma$ (spread).
5. Empirical rule: about $68\%$ of values lie within $\mu\pm\sigma$, $95\%$ within $\mu\pm 2\sigma$, and $99.7\%$ within $\mu\pm 3\sigma$.
6. Quartiles: $Q_1=\mu-0.6745\sigma$, $Q_3=\mu+0.6745\sigma$; it has zero skewness and kurtosis $3$.

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

Given $\mu = 50$, $\sigma = 10$. Standardize using $Z = \dfrac{X-\mu}{\sigma}$.

For $X = 45$:

For $X = 62$:

So we need $P(-0.5 < Z < 1.2)$. From the standard normal table:

• $P(0 < Z < 0.5) = 0.1915$
• $P(0 < Z < 1.2) = 0.3849$

The probability that a value lies between 45 and 62 is $0.5764$ (about $57.64\%$).

Hypothesis Testing

Hypothesis testing is a statistical procedure used to decide, on the basis of sample evidence, whether a stated assumption (hypothesis) about a population parameter should be accepted or rejected at a chosen level of significance.

Procedure

1. Set up the hypotheses

• Null hypothesis ($H_0$): a statement of no difference / no effect, assumed true unless evidence contradicts it, e.g. $H_0:\mu = \mu_0$.
• Alternative hypothesis ($H_1$): the claim accepted if $H_0$ is rejected. It may be two-tailed ($\mu \ne \mu_0$) or one-tailed ($\mu > \mu_0$ or $\mu < \mu_0$).

2. Choose the level of significance ($\alpha$) The probability of rejecting $H_0$ when it is actually true. Common values are $\alpha = 0.05$ ($5\%$) or $\alpha = 0.01$ ($1\%$).

3. Identify the types of error

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

4. Select the test statistic appropriate to the problem (e.g. $Z$, $t$, $\chi^2$, $F$) and compute its value from the sample.

5. Determine the critical region The critical (rejection) region is the set of values of the test statistic for which $H_0$ is rejected; its size equals $\alpha$. The boundary value is the critical value. For a two-tailed test the region lies in both tails; for a one-tailed test in a single tail.

6. Make the decision If the computed test statistic falls in the critical region (i.e. $|\text{calculated}| > \text{tabulated value}$), reject $H_0$; otherwise do not reject $H_0$. State the conclusion in terms of the original 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, $P(\text{King or Queen}) = \frac{4}{52} + \frac{4}{52} = \frac{8}{52} = \frac{2}{13}$ (the events are mutually exclusive).

Multiplication Theorem of Probability

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

If $A$ and $B$ are independent, then $P(B\mid A)=P(B)$ and:

Example: Tossing a fair coin and rolling a fair die, $P(\text{Head and } 6) = \frac{1}{2}\times\frac{1}{6} = \frac{1}{12}$ (independent events).

Poisson Distribution

The Poisson distribution gives the probability of a given number of independent events occurring in a fixed interval of time or space, when these events occur with a known constant average rate $\lambda$ and independently of the time since the last event.

Its probability mass function is:

where $\lambda > 0$ is the average number of occurrences. It is the limiting case 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 $(=\lambda)$.

Applications

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 defective items in a large batch (rare-event quality control).
4. Number of customers arriving at a service counter per unit time.
5. Number of radioactive particles emitted per second; modelling arrivals in queueing/network traffic.

Random Variable

A random variable is a real-valued function $X$ that assigns a numerical value to each outcome (sample point) of a random experiment. It maps the sample space $S$ to the set of real numbers $\mathbb{R}$.

Example: In tossing two coins, the number of heads $X \in \{0,1,2\}$ is a random variable.

Discrete vs Continuous Random Variables

In short: a discrete random variable arises from counting, whereas a continuous random variable arises from measurement.

Mathematical Expectation

The mathematical expectation (or mean) of a random variable $X$ is the long-run average value, weighted by probabilities.

• Discrete: $E(X) = \sum_x x\,P(X=x)$
• Continuous: $E(X) = \int_{-\infty}^{\infty} x\,f(x)\,dx$

Properties (with proofs)

1. Expectation of a constant. $E(c) = c$.

Proof: $E(c) = \sum_x c\,P(X=x) = c\sum_x P(X=x) = c\cdot 1 = c.$

2. Linearity (constant multiplier). $E(aX) = a\,E(X)$.

Proof: $E(aX) = \sum_x a x\,P(X=x) = a\sum_x x\,P(X=x) = a\,E(X).$

3. Addition of a constant. $E(aX + b) = a\,E(X) + b$.

Proof: $E(aX+b)=\sum_x (ax+b)P(x)=a\sum_x xP(x)+b\sum_x P(x)=aE(X)+b.$

4. Addition theorem. $E(X + Y) = E(X) + E(Y)$ for any two random variables.

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

5. Multiplication theorem (independence). If $X$ and $Y$ are independent, $E(XY)=E(X)\,E(Y)$.

Proof: $E(XY)=\sum_x\sum_y xy\,P(x)P(y)=\Big(\sum_x xP(x)\Big)\Big(\sum_y yP(y)\Big)=E(X)E(Y).$

t-test for Difference of Two Sample Means

This test is used to determine whether the difference between the means of two independent small samples ($n_1, n_2 < 30$) drawn from normal populations with equal (unknown) variances is statistically significant.

Hypotheses

• $H_0: \mu_1 = \mu_2$ (the two population means are equal)
• $H_1: \mu_1 \ne \mu_2$ (two-tailed) or one-sided as required

Test Statistic

where the pooled standard deviation is:

Degrees of Freedom

Decision Rule

Compare $|t_{\text{cal}}|$ with the tabulated value $t_{\alpha}$ at the given degrees of freedom and level of significance $\alpha$:

• If $|t_{\text{cal}}| > t_{\text{tab}}$, reject $H_0$ — the difference is significant.
• If $|t_{\text{cal}}| \le t_{\text{tab}}$, accept $H_0$ — the difference is not significant.

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

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, and the z-test is used to test whether the sample mean $\bar{x}$ differs significantly from a hypothesized population mean $\mu$.

Hypotheses

• $H_0: \mu = \mu_0$
• $H_1: \mu \ne \mu_0$ (two-tailed), at level of significance $\alpha$.

Test Statistic

If the population standard deviation $\sigma$ is unknown, the sample standard deviation $s$ is used in its place (valid for large $n$).

Decision Rule

At $\alpha = 0.05$ (two-tailed) the critical value is $1.96$. If $|Z_{\text{cal}}| > 1.96$, reject $H_0$; otherwise accept $H_0$.

Example

A sample of $n = 64$ items has mean $\bar{x} = 52$ from a population with $\mu = 50$ and $\sigma = 8$. Test at $5\%$ level.

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

Karl Pearson's Coefficient of Correlation

Karl Pearson's coefficient of correlation $r$ is a numerical measure of the degree and direction of linear relationship between two variables $X$ and $Y$. It is defined as the ratio of the covariance of $X$ and $Y$ to the product of their standard deviations:

Properties

1. Range: $-1 \le r \le +1$.
   • $r = +1$: perfect positive correlation; $r = -1$: perfect negative correlation; $r = 0$: no linear correlation.
2. Independent of origin and scale: $r$ is unchanged if a constant is added to or each value is multiplied by a constant.
3. Symmetric: $r_{xy} = r_{yx}$.
4. It is the geometric mean of the two regression coefficients: $r = \pm\sqrt{b_{xy}\cdot b_{yx}}$, taking the sign of the regression coefficients.
5. It measures only the linear relationship; $r=0$ does not imply the variables are independent (a nonlinear relation may exist).

Regression Coefficients

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

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

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

Properties

1. Correlation as geometric mean: $r = \pm\sqrt{b_{yx}\cdot b_{xy}}$, so $r^2 = b_{yx}\,b_{xy}$.
2. Same sign: both regression coefficients have the same sign as $r$; if one is positive both are positive, and vice versa.
3. Since $|r|\le 1$, the product $b_{yx}\,b_{xy} \le 1$ (both cannot exceed $1$ simultaneously).
4. They are 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. $\dfrac{b_{yx}+b_{xy}}{2} \ge r$.

Sampling Distribution

A sampling distribution is the probability distribution of a sample statistic (such as the sample mean $\bar{x}$ or proportion $p$) obtained from all possible samples of the same size $n$ drawn from a given population. For example, if repeated random samples of size $n$ are taken and the mean of each is computed, the distribution of those means is the sampling distribution of the mean.

By the Central Limit Theorem, for large $n$ the sampling distribution of the mean is approximately normal with mean $\mu$ (the population mean) and standard deviation $\sigma/\sqrt{n}$, regardless of the shape of the parent population.

Standard Error

The standard error (S.E.) is the standard deviation of the sampling distribution of a statistic. It measures the variability of the statistic from sample to sample.

• S.E. of the mean: $\quad \text{S.E.}(\bar{x}) = \dfrac{\sigma}{\sqrt{n}}$
• S.E. of a proportion: $\quad \text{S.E.}(p) = \sqrt{\dfrac{PQ}{n}}$

Significance: a smaller standard error indicates greater precision (estimates cluster tightly around the parameter). The S.E. decreases as the sample size $n$ increases, and it is used to construct confidence intervals and test statistics in tests of significance.