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Let f : [0, 1] → [0, 1] be the modified Dirichlet function defined as f(x) = if c = " is rational in lowest terms, 0 if r is irrational and let h : [0, 1] → [0, 1] be the function 1 if x is rational, h(x) = 0 if x is irrational Find an integrable function g: [0,1] [0, 1] such that h=gof, thereby showing that the composition of two integrable functions need not be integrable.
A tradesman marks his goods 25% above the cost price and allows his customers 12% reduction on their bills. What percent profit does he make?
Let R1 and R2 be two relation on real number such that R1 = {(x, y)|x < y} and R2 =
{(x, y)|x > y}, then find R1 ∪ R2,R1 ∩ R2,R1 − R2, R2 − R1, and R1+LR2.
There are 150 students in a class. The distribution if their marks in a mathematics test are as follows
Class frequency
0-9 3
10-19 10
20-29 17
30-39 x
40-49 35
50-59 y
60-69 18
70-79 10
80-89 5
90-99 2
Required
i) The value of x given that the median mark is 44.357 (2marks)
ii) The value of y given that the modal is 43.0 (2marks)
iii) Draw an ogive of the data in (a) above (3 marks)
A real estate agent sells two sites for P18,000 each. On one he gains 25% and on the other he loses 25%. What is his loss or gain percent?
Suppose that the average outstanding credit card balance for young professionals
is Php11,200 with standard deviation of Php 2,600. In simple random sample of
150 young professionals, what is the probability that the mean outstanding credit
card balance does not exceeds Php12,300?
QUESTION THREE
In a Competitive examination of 5000 students, the marks of the examinees in statistics were found to be distributed normally with mean 45 and standard deviations 14.
Determine the number of examinees whose marks, out of 100 were;
(i) Less than 30. 2MKS
(ii) Between 30 and 70. 2MKS
(iii) Between 60 and 80. 2MKS
(iv) More than 60. 2MKS
(v) More than 40 2MKS
Consider all the possible samples of size 2 that can be drawn without replacement from the population 1, 4, 6 and 7.
Compute the following;
a. Mean of the sampling distribution of the sample means
b. Variance of the sampling distribution of the sample means
c. Standard deviation of the sampling distribution of sample means
An individual is handover a bag filled with balls of different colours. The table below
gives the probability that a randomly chosen ball is of a specific colour, along with a
missing probability value of picking a ball of white colour.
Colour Red Yellow Green Orange Blue White
Probability 0.2 0.2 0.1 0.1 0.3 ?
Answer the following questions:
(i) What value must the missing probability be?
(ii) If a ball is drawn at random from the bag, what is the probability of each of the
following events?
I. It is either Blue or Red in colour.
II. It is notYellow in colour.
III. It is neither Orange nor White in colour.
IV. It is either Blue or Red or Yellow or Green or Orange or White.
Consider a Cobb-Douglas production function Q =ALαKβ where the amount produced (Q)
is given as a function of the labor (L) and capital (K) used and A, α, β are constants. Let
the value of the constants in the above production function be given by A = 1, α = 2/3
, β = 1/3
.
Wage rate (w) and per unit capital rate (r) be Rs 4 and Rs 27, respectively. Suppose that
the firm wishes to produce 1080 units of output Q. What will be the optimal amount of
factors that the firm needs to employ for this? Also calculate the minimum cost of
producing such an output level?
#340153 on Dec 2023