(a) Compute (i) Laspeyre's (ii) Paasche's (iii) Fisher's Index numbers for the year 2010 from the following data. | Commodity | Price 2000 | Price 2010 | Quantity 2000 | Quantity 2010 | | --- | --- | --- | --- | --- | | A | 12 | 14 | 18 | 16 | | B | 15 | 16 | 20 | 15 | | C | 14 | 15 | 24 | 20 | | D | 12 | 12 | 29 | 23 | OR (b) The probability function of a random variable X is given by $p(x)=\begin{cases}\dfrac{1}{4}, & \text{for } x=-2 \\ \dfrac{1}{4}, & \text{for } x=0 \\ \dfrac{1}{2}, & \text{for } x=10 \\ 0, & \text{elsewhere}\end{cases}$ Evaluate the following probabilities (i) $P(X\le 0)$ (ii) $P(X<0)$ (iii) $P(|X|\le 2)$ (iv) $P(0\le X\le 10)$

> > [!TLDR] > (a) $L=106.67,\ P=106.89,\ F=106.78$. (b) $\tfrac12,\ \tfrac14,\ \tfrac12,\ \tfrac34$. > > **(a) Index numbers for 2010.** Compute the products: > > | Commodity | $p_0$ | $p_1$ | $q_0$ | $q_1$ | $p_1q_0$ | $p_0q_0$ | $p_1q_1$ | $p_0q_1$ | > | --- | --- | --- | --- | --- | --- | --- | --- | --- | > | A | 12 | 14 | 18 | 16 | 252 | 216 | 224 | 192 | > | B | 15 | 16 | 20 | 15 | 320 | 300 | 240 | 225 | > | C | 14 | 15 | 24 | 20 | 360 | 336 | 300 | 280 | > | D | 12 | 12 | 29 | 23 | 348 | 348 | 276 | 276 | > | **Total** | | | | | **1280** | **1200** | **1040** | **973** | > > $$\text{Laspeyre's } L=\frac{\sum p_1q_0}{\sum p_0q_0}\times100=\frac{1280}{1200}\times100=106.67.$$ > $$\text{Paasche's } P=\frac{\sum p_1q_1}{\sum p_0q_1}\times100=\frac{1040}{973}\times100=106.89.$$ > $$\text{Fisher's } F=\sqrt{L\times P}=\sqrt{106.67\times106.89}=106.78.$$ > …

(a) Compute (i) Laspeyre's (ii) Paasche's (iii) Fisher's Index numbers for the year 2010 from the following da

Question 26 of 39

(a) Compute (i) Laspeyre's (ii) Paasche's (iii) Fisher's Index numbers for the year 2010 from the following data.

Commodity	Price 2000	Price 2010	Quantity 2000	Quantity 2010
A	12	14	18	16
B	15	16	20	15
C	14	15	24	20
D	12	12	29	23

(b) The probability function of a random variable X is given by $p(x)=\begin{cases}\dfrac{1}{4}, & \text{for } x=-2 \\ \dfrac{1}{4}, & \text{for } x=0 \\ \dfrac{1}{2}, & \text{for } x=10 \\ 0, & \text{elsewhere}\end{cases}$

Evaluate the following probabilities

(i) $P(X\le 0)$ (ii) $P(X<0)$ (iii) $P(|X|\le 2)$ (iv) $P(0\le X\le 10)$

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(a) $L=106.67,\ P=106.89,\ F=106.78$ . (b) $\tfrac12,\ \tfrac14,\ \tfrac12,\ \tfrac34$ .

(a) Index numbers for 2010. Compute the products:

Commodity	$p_0$	$p_1$	$q_0$	$q_1$	$p_1q_0$	$p_0q_0$	$p_1q_1$	$p_0q_1$
A	12	14	18	16	252	216	224	192
B	15	16	20	15	320	300	240	225
C	14	15	24	20	360	336	300	280
D	12	12	29	23	348	348	276	276
Total					1280	1200	1040	973

$\text{Laspeyre's } L=\frac{\sum p_1q_0}{\sum p_0q_0}\times100=\frac{1280}{1200}\times100=106.67.$

$\text{Paasche's } P=\frac{\sum p_1q_1}{\sum p_0q_1}\times100=\frac{1040}{973}\times100=106.89.$

$\text{Fisher's } F=\sqrt{L\times P}=\sqrt{106.67\times106.89}=106.78.$

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