BSc CSIT (TU) Science Statistics I (BSc CSIT, STA164) Question Paper 2080 Nepal

Measure of Dispersion

A measure of dispersion (or variation) describes how much the individual observations in a data set are spread out or scattered about a central value (mean, median). While a measure of central tendency gives a single representative value, a measure of dispersion tells us how reliable that value is. Small dispersion means the data are homogeneous and clustered; large dispersion means they are heterogeneous and widely scattered.

Common measures: Range, Quartile Deviation (Q.D.), Mean Deviation (M.D.) and Standard Deviation (S.D.). Range and Q.D. are positional (based on quartiles); M.D. and S.D. are calculation-based (based on deviations from an average).

Worked Example

Take the data: $5, 7, 9, 11, 13, 15, 17$ (n = 7), already arranged in order.

Mean $\bar{x} = \dfrac{5+7+9+11+13+15+17}{7} = \dfrac{77}{7} = 11$.

(a) Quartile Deviation

Position of $Q_1 = \left(\dfrac{n+1}{4}\right)^{th} = 2^{nd}$ item $= 7$.

Position of $Q_3 = \left(\dfrac{3(n+1)}{4}\right)^{th} = 6^{th}$ item $= 15$.

(b) Mean Deviation (about the mean)

$\sum|x-\bar{x}| = 24$, so $M.D. = \dfrac{24}{7} = 3.43$.

(c) Standard Deviation

$\sum(x-\bar{x})^2 = 36+16+4+0+4+16+36 = 112$.

Comparison

For this symmetric data the values are close. In general, S.D. is the most reliable measure because it uses every observation, is amenable to algebraic treatment and is the basis of further statistical analysis; Q.D. is preferred for open-ended distributions, and M.D. is simple but less suitable for further computation (theoretically $M.D. < S.D.$ for most distributions).

Conditional Probability

The conditional probability of event $A$ given that event $B$ has already occurred is

It re-scales the sample space to those outcomes in which $B$ occurs.

Independent Events

Two events $A$ and $B$ are independent if the occurrence of one does not affect the probability of the other:

Bayes' Theorem

If $B_1, B_2, \dots, B_n$ are mutually exclusive and exhaustive events with $P(B_i) > 0$, and $A$ is any event with $P(A) > 0$, then

The $P(B_i)$ are prior probabilities and $P(B_i\mid A)$ are posterior probabilities; the denominator is $P(A)$ by the law of total probability.

Applied Problem

Three machines $B_1, B_2, B_3$ produce 50%, 30% and 20% of a factory's output, with defective rates 3%, 4% and 5% respectively. An item is found defective; what is the probability it came from machine $B_3$?

Priors: $P(B_1)=0.5,\ P(B_2)=0.3,\ P(B_3)=0.2$. Likelihoods (defective $=A$): $P(A\mid B_1)=0.03,\ P(A\mid B_2)=0.04,\ P(A\mid B_3)=0.05$.

Total probability of a defective item:

By Bayes' theorem:

Conclusion: Given the item is defective, there is about a 27% chance it was produced by machine $B_3$.

Mathematical Expectation

The mathematical expectation (expected value) of a random variable is its long-run average value, weighting each value by its probability. For a discrete random variable $X$ taking values $x_i$ with probabilities $p_i = P(X=x_i)$,

Mean and Variance from a Distribution

Consider the distribution:

Mean:

For variance compute $E(X^2)$:

Standard deviation $\sigma = \sqrt{0.81} = 0.9$.

Properties of Expectation

1. $E(c) = c$ for any constant $c$.
2. $E(cX) = c\,E(X)$.
3. $E(aX + b) = a\,E(X) + b$ (linearity).
4. $E(X + Y) = E(X) + E(Y)$ — addition theorem (always holds).
5. If $X$ and $Y$ are independent, $E(XY) = E(X)\,E(Y)$ — multiplication theorem.
6. $Var(X) = E(X^2) - [E(X)]^2$, and $Var(aX+b) = a^2 Var(X)$.

The mathematical expectation (or expected value) of a random variable $X$ is its probability-weighted average value, denoted $E(X)$ or $\mu$.

• Discrete: $E(X) = \sum_i x_i\, p_i$, where $p_i = P(X = x_i)$ and $\sum p_i = 1$.
• Continuous: $E(X) = \displaystyle\int_{-\infty}^{\infty} x\, f(x)\, dx$, where $f(x)$ is the probability density function.

It represents the long-run average value of $X$ over many repetitions of the experiment and serves as the theoretical mean of the distribution.

Bayes' theorem relates posterior probability to prior probability and likelihood. If $B_1, B_2, \dots, B_n$ are mutually exclusive and exhaustive events (a partition of the sample space) with $P(B_i) > 0$, and $A$ is any event with $P(A) > 0$, then

Here $P(B_i)$ are the prior probabilities, $P(A\mid B_i)$ the likelihoods, and $P(B_i\mid A)$ the posterior probabilities. The denominator equals $P(A)$ by the law of total probability.

A parameter is a numerical characteristic of the entire population, whereas a statistic is a numerical characteristic computed from a sample.

In short, a statistic (e.g. sample mean $\bar{x}$) is used to estimate the corresponding unknown parameter (e.g. population mean $\mu$).

The variance of a random variable $X$ measures the average squared deviation of $X$ from its mean $\mu = E(X)$. It quantifies the spread of the distribution:

• Discrete: $Var(X) = \sum_i (x_i - \mu)^2 p_i$.
• Continuous: $Var(X) = \displaystyle\int_{-\infty}^{\infty} (x-\mu)^2 f(x)\, dx$.

Variance is always non-negative, and its positive square root $\sigma = \sqrt{Var(X)}$ is the standard deviation. Also $Var(aX+b) = a^2\,Var(X)$.

A probability distribution of a random variable is a description that assigns probabilities to all the possible values the variable can take, listing each value together with its probability of occurrence.

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

Example (discrete): For a fair die, $P(X=x) = \frac{1}{6}$ for $x = 1,2,\dots,6$. Examples include the Binomial, Poisson and Normal distributions.

The weighted arithmetic mean is an average in which each value $x_i$ is given a weight $w_i$ reflecting its relative importance, instead of treating all observations equally. It is defined as

It is used when the items do not contribute equally (e.g. computing a GPA where credit hours are weights, or an index number). When all weights are equal it reduces to the ordinary arithmetic mean.

Example: Marks 80, 70, 90 with weights (credits) 3, 2, 5: $\bar{x}_w = \dfrac{3(80)+2(70)+5(90)}{3+2+5} = \dfrac{830}{10} = 83$.

Properties of mathematical expectation:

1. Constant: $E(c) = c$ for any constant $c$.
2. Scalar multiple: $E(cX) = c\,E(X)$.
3. Linearity: $E(aX + b) = a\,E(X) + b$.
4. Addition theorem: $E(X + Y) = E(X) + E(Y)$ (holds for any $X, Y$).
5. Multiplication theorem: if $X$ and $Y$ are independent, $E(XY) = E(X)\,E(Y)$.
6. $E(X)$ lies between the minimum and maximum values of $X$; and if $X \ge 0$ then $E(X) \ge 0$.
7. Relation to variance: $Var(X) = E(X^2) - [E(X)]^2$.

The interquartile range (IQR) is the difference between the third quartile $Q_3$ (75th percentile) and the first quartile $Q_1$ (25th percentile):

It is a measure of dispersion covering the middle 50% of the data and is not affected by extreme values (outliers), making it a robust measure of spread. The related quartile deviation (semi-interquartile range) is $\dfrac{Q_3 - Q_1}{2}$.

Example: If $Q_1 = 25$ and $Q_3 = 45$, then $IQR = 45 - 25 = 20$.

When a fair coin is tossed once, the sample space is $S = \{H, T\}$ with $n(S) = 2$ equally likely outcomes. The favourable event $A = \{H\}$ has $n(A) = 1$.

Thus the probability of getting a head is $\tfrac{1}{2}$ or 0.5 (50%).