Statistics

Statistics of birth and death

The statistics of birth and death involve vital demographic indices such as crude birth and death rates, age-specific fertility rates, and mortality tables, which are fundamental for population projection.

For: UP PGT Mathematics UP TGT Mathematics DSSSB PGT / TGT KVS / NVS PGT LT Grade

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Definition and statement

Demographic Vital Rates and Force of Mortality Principle

Let $N(t)$ denote the population size at time $t$, $B(t)$ the number of births, and $D(t)$ the number of deaths in $[t, t+dt)$. The force of mortality $\mu_x$ at age $x$ is defined as the instantaneous rate of mortality per unit of population at exact age $x$, given by $l_x \mu_x = -\frac{d}{dx} l_x$, where $l_x$ represents the number of survivors at age $x$ out of an initial cohort $l_0$.

📐 Geometric Interpretation

The survivorship curve $l_x$ versus age $x$ represents a strictly non-increasing function starting at $l_0$ and terminating at $l_w = 0$, where $w$ is the limiting age. The area under the survivorship curve yields the total number of person-years lived by the cohort.

⚙️ Algebraic & Structural Meaning

Establishes differential equation models for population dynamics and actuarial mathematics, connecting probability density functions of human lifetime with empirical mortality tables.

Conditions and hypotheses

Check each condition and what changes when it is omitted:

Hypothesis / Condition	Domain / Interval	Necessity	What Fails If Condition Dropped
Survivorship Continuity and Differentiability	`[0, w] (Cohort lifespan interval)`	Strictly Mandatory / Necessary	Discontinuities in $l_x$ imply instantaneous catastrophic cohort losses which violate standard demographic smoothness assumptions.
Non-negativity of Force of Mortality	`\mu_x \ge 0 \quad \forall x \in [0, w]`	Strictly Mandatory	Negative mortality rates violate probability axioms and imply population resurrection.
Initial Cohort Normalization	`l_0 > 0`	Essential for conclusion	If $l_0 = 0$, the entire cohort analysis is trivialized as a null set.

Useful results and shortcuts

Use these identities and results when their conditions apply:

Probability of Survival

$$ _t p_x = \frac{l_{x+t}}{l_x} = \exp\left(-\int_0^t \mu_{x+s} ds\right) $$

Rule: Instantly convert cumulative survivorship ratios into exponential integrals of the force of mortality.

Application: Frequently asked in UP PGT actuarial-statistics sections for rapid calculation.

Crude Death Rate (CDR) Formula

$$ \text{CDR} = \frac{D}{P} \times 1000 $$

Rule: Direct ratio of total deaths $D$ to mid-year total population $P$, expressed per thousand.

General Fertility Rate (GFR)

$$ \text{GFR} = \frac{\text{Total Births}}{\text{Female Population aged 15-49}} \times 1000 $$

Rule: Always check denominator bounds: strictly women in childbearing age (15-49 years).

Application: Direct evaluation of demographic definitions.

Net Reproduction Rate (NRR) Threshold

$$ \text{NRR} = \int_{15}^{49} l_a \cdot m_a \cdot \frac{L_a}{l_0} da $$

Rule: If $\text{NRR} > 1$, population grows; if $\text{NRR} < 1$, population declines; if $\text{NRR} = 1$, replacement level.

Application: Conceptual trick questions testing demographic equilibrium.

Expectation of Life ($e_x$)

$$ e_x = \frac{1}{l_x} \int_0^{w-x} l_{x+t} dt = \sum_{k=0}^{\infty} {}_k p_x $$

Rule: Average remaining lifetime is equal to the integral of future survival probabilities.

Application: Direct calculation in numerical problems.

Rapid Elimination via Units

$$ \text{Rates per } 1000 \implies \text{Multiply raw decimal fractions by } 1000 $$

Rule: Eliminate options failing magnitude or scaling checks instantly.

Common errors and counterexamples

These examples show how a conclusion can fail when a condition is missing:

⚠️ COMMON ERROR #01

CDR vs STDR Confusion

Common Pitfall: Using Crude Death Rate to compare mortality across countries with different age structures without age-adjustment (Standardized Death Rate).

Counterexample: Country A (older population) having higher CDR than Country B despite better healthcare.

Why it fails: CDR is confounded by age distribution; direct comparison is invalid.

🛡️ Exam Defense: Always look for age-standardization when comparing populations.

⚠️ COMMON ERROR #02

GFR Denominator Misidentification

Common Pitfall: Including total population or males in the denominator of General Fertility Rate.

Counterexample: Dividing total annual births by total mid-year population.

Why it fails: This yields Crude Birth Rate, not General Fertility Rate.

🛡️ Exam Defense: GFR denominator must exclusively comprise women aged 15-49.

Comparison

Concept / Theorem	Hypotheses / Assumptions	Key Formula / Outcome	Geometric Meaning	Exam Identification Clue
Crude Birth Rate (CBR)	Total population denominator	$CBR = \frac{B}{P} \times 1000$	Overall population addition intensity	Keywords: 'total births', 'mid-year population'
General Fertility Rate (GFR)	Women aged 15-49 denominator	$GFR = \frac{B}{F_{15-49}} \times 1000$	Reproductive performance index	Keywords: 'women of childbearing age'
Infant Mortality Rate (IMR)	Live births denominator	$IMR = \frac{D_{<1 \text{ yr}}}{\text{Live Births}} \times 1000$	Early childhood survival vulnerability	Keywords: 'deaths under one year'
Net Reproduction Rate (NRR)	Cohort survival & female births	$NRR = \sum l_x m_x \dots$	Generational replacement factor ($>1$ growth)	Keywords: 'replacement level', 'daughters per woman'

Solved examples

Each problem includes a full solution and a shorter method:

CANONICAL PROBLEM 01 Dual-Method Breakdown

In a population, the total number of live births in a year is $15,000$ and the female population in the age group 15-49 years is $60,000$. Find the General Fertility Rate (GFR).

(A) $125$(B) $250$(C) $300$(D) $150$

📚 Method A: Formal Textbook Derivation

Step 1: Identify formula $GFR = \frac{\text{Births}}{\text{Female Pop (15-49)}} \times 1000$. Step 2: Substitute values: $\frac{15000}{60000} \times 1000 = \frac{1}{4} \times 1000 = 250$.

⚡ Method B: 25-Second Exam Shortcut

Compute ratio $\frac{15}{60} = \frac{1}{4}$, multiply by 1000 to get 250.

💡 Tip: Verify denominator is strictly 15-49 female population, not total population.

CANONICAL PROBLEM 02 Dual-Method Breakdown

If the force of mortality $\mu_x$ is constant and equal to $m$, what is the probability that a person aged $x$ will survive for $t$ years?

(A) $e^{-mt}$(B) $1 - e^{-mt}$(C) $mt$(D) $e^{mt}$

📚 Method A: Formal Textbook Derivation

Step 1: Use $_t p_x = \exp\left(-\int_0^t \mu_{x+s} ds\right)$. Step 2: Since $\mu_{x+s} = m$, the integral evaluates to $\int_0^t m ds = mt$. Step 3: Result is $e^{-mt}$.

⚡ Method B: 25-Second Exam Shortcut

Constant force of mortality directly leads to exponential survival law $e^{-mt}$.

💡 Tip: Remember constant force analogy with radioactive decay.

Practice questions

Choose an answer to check it and see the solution:

MCQ 1: The Crude Death Rate (CDR) is expressed per:

A 100

B 1,000

C 10,000

D 1,000,000

Derivation & Analysis:

By definition, CDR is calculated per thousand of the mid-year population: $\frac{D}{P} \times 1000$.

MCQ 2: If $l_0 = 100000$ and $l_x$ is the number of survivors at age $x$, the probability of dying between age $x$ and $x+1$ is given by:

A $\frac{l_x - l_{x+1}}{l_x}$

B $\frac{l_{x+1}}{l_x}$

C $\frac{l_x}{l_{x+1}}$

D $l_x - l_{x+1}$

Derivation & Analysis:

Probability of dying in the interval is the ratio of deaths in $[x, x+1]$, which is $l_x - l_{x+1}$, divided by the cohort size $l_x$.

MCQ 3: General Fertility Rate (GFR) uses which of the following as its denominator?

A Total population

B Total female population

C Female population aged 15-49 years

D Total male population

Derivation & Analysis:

GFR is specifically restricted to women in the reproductive age span of 15 to 49 years.

MCQ 4: Infant Mortality Rate (IMR) has which denominator?

A Total population

B Total live births

C Total deaths

D Female population

Derivation & Analysis:

IMR is defined as $\frac{\text{Deaths of infants under 1 year}}{\text{Total live births}} \times 1000$.

MCQ 5: If Net Reproduction Rate (NRR) is equal to 1, then the population is:

A Increasing

B Decreasing

C Stationary / Replacement level

D Extinct

Derivation & Analysis:

An NRR of exactly 1 means each generation of mothers replaces itself exactly, leading to zero long-term population growth.

MCQ 6: The force of mortality $\mu_x$ in terms of survivorship $l_x$ is expressed as:

A $-\frac{1}{l_x} \frac{dl_x}{dx}$

B $\frac{1}{l_x} \frac{dl_x}{dx}$

C $-\frac{dl_x}{dx}$

D $\frac{dl_x}{dx}$

Derivation & Analysis:

Since $l_x \mu_x = -\frac{dl_x}{dx}$, dividing by $l_x$ yields $\mu_x = -\frac{1}{l_x} \frac{dl_x}{dx}$.

MCQ 7: Which demographic measure is least affected by age structure variations?

A Crude Birth Rate

B Crude Death Rate

C Age-Specific Death Rate

D Total Population Growth Rate

Derivation & Analysis:

Age-Specific Death Rates (ASDR) isolate specific age cohorts, eliminating age distribution confounding effects.

MCQ 8: If survivorship function $l_x = 100 - x$ for $0 \le x \le 100$, what is the limiting age $w$?

A 50

B 100

C 75

D Infinity

Derivation & Analysis:

Setting $l_w = 100 - w = 0$ gives $w = 100$.

MCQ 9: The expectation of life at birth $e_0$ is given by:

A $\int_0^w l_x dx$

B $\frac{1}{l_0} \int_0^w l_x dx$

C $\int_0^w x l_x dx$

D $\frac{1}{l_0} \int_0^w x l_x dx$

Derivation & Analysis:

Expectation of life is total person-years lived divided by initial cohort $l_0$, i.e., $\frac{1}{l_0} \int_0^w l_x dx$.

Q10

MCQ 10: In vital statistics, a life table column denoted by $L_x$ represents:

A Number of survivors at age $x$

B Number of person-years lived between age $x$ and $x+1$

C Deaths between age $x$ and $x+1$

D Force of mortality

Derivation & Analysis:

Standard actuarial notation $L_x = \int_0^1 l_{x+t} dt$ represents person-years lived in the age interval $[x, x+1]$.PILLAR_7_COMPLETE

Common questions

Multiplication by 1000 converts small decimal fractions into convenient whole numbers for demographic comparison and reporting.

CBR uses the entire total population as the denominator, whereas GFR restricts the denominator strictly to women of childbearing age (15-49 years).

Memorize standard integral definitions and recognize exponential survival relations to bypass lengthy derivations.

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