Maths JNU

[Chandrani Sarma]

A biased dice is rolled independently three times with the probability of an even number appearing being twice that of an odd number. We define two random variables Xi and Y.
Xi = 1, if the i th roll gives an odd number
   =  0, otherwise

Y = X1+ X2+ X3

(a) Find the probability distribution of Xi
(b) Find the probability distribution of Y
(c) What are the mean and variance of Y?

[Amit Goyal]

Hi Chandrani,
Since dice is independently rolled three times, probability distribution of Xi is:
P(Xi = 0) = P(Even number on ith roll) = 2/3
P(Xi = 1) = P(Odd number on ith roll) = 1/3

Probability distribution of Y is
P(Y = 0) = P(X1 = X2 = X3 = 0) = 8/27
P(Y = 1) = P(X1 = X2 = 0, X3 = 1) + P(X1 = X3 = 0, X2 = 1) + P(X2 = X3 = 0, X1 = 1) = 12/27
P(Y = 2) = P(X1 = X2 = 1, X3 = 0) + P(X1 = X3 = 1, X2 = 0) + P(X2 = X3 = 1, X1 = 0) = 6/27
P(Y = 3) = P(X1 = X2 = X3 = 1) = 1/27

Mean(Y) = 1 Variance(Y) = 2/3

[Chandrani Sarma]

Thank you, Sir. :)

[Chandrani Sarma]

In 1990, workers obtained a wage of Rs.1000 and were observed to spend 40% on 'Food', 30% each on 'Clothing' and 'other items'. By 1996, the wage was Rs 1500 but during the period, food prices increased by 100%, cloth prices by 50% and prices of other items by 10%.
What is the percentage change in real wages?

[duck]

In 1990, wage =1000
as per given , expenditure on food =400 , exp on cloth= 300 and exp on other items= 300

now , if he buys the same bundle in 1996 then,
exp on food=800 , exp on cloth= 450 , exp on other items= 330
total exp = 1580 > 1500 , income in 1996
so there is a fall in real income as he cannot afford at today's income what he could buy yesterday..

therefore , % change in real wage= 1500-1580
                                                ----------  * 100
                                                   1580

                                             = -8000
                                                ------  
                                                 1580

                                            = -5.06%

[Amit Goyal]

That's right Nidhi.

[ritu]

hello sir....
rate of growth of wage is (1500-1000)/1000*100=50%
the cost of same bundle is 1580 so inflation is (1580-1000)/1000*100=58%

now if i take Y=w/p
take log on both sides logy=log w - log p
diff both sides wrt "t"...Gy=Gw-Gp
=50-58
=-8%


what is wrong with this.     :(((((    ???????

[anon_econ]

hey ritu i guess v shud avoid calculus approximations when working with actual numbers..for eg in jnu 2011 question 2 if u use the rate of growth formula u'll get 40% but that's an approximation..the actual answer is 44%