MCA4020 - PROBABILITY AND STATISTICS

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SUMMER 2015 ASSIGNMENT

DRIVE	SUMMER 2015
PROGRAM	MCA
SEMESTER	FOURTH
SUBJECT CODE & NAME	MCA4020- PROBABILITY AND STATISTICS
BK ID	B1779
CREDIT & MARKS	4 CREDITS & 60 MARKS

1 Three machines A, B and C produce respectively 60%, 30% and 10% of the total number of items of a factory. The percentage of defective output of these machines are respectively 2%, 3% and 4%. An item is selected at random and is found to be defective. Find the probability that the item was produced by machine C.

Answer: Multiply the respective percentage of production by the percentage of defects for each machine

A = 60% * 2% = 1.2% of the factory's total output are defects produced by Machine A

B = 30% * 3% = 0.9% of the factory's total output are defects produced by Machine B

C = 10% * 4% = 0.4% of the factory's total output are defects produced by Machine C

Therefore the total defects = 1.2% + 0.9% + 0.4% = 2.5%

2 Find the constant k so that

F(x,y) = {k(x+1)e^-y, 0<x<1,y>0

                0 elsewhere

Is a joint probability density function. Are X and Y independent?

Answer: Probability density function:- In probability theory and statistics, given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value x of X as a parameter. In case that both "X" and "Y" are categorical variables, a conditional probability table is typically used to represent the conditional probability. The conditional distribution

3 The data shows the distribution of weight of students of 1st standard of a school. Find the quartiles.

Class Interval	13-18	18-20	20-21	21-22	22-23	23-25	25-30
Frequency	22	27	51	42	32	16	10

Answer: If a data set of scores is arranged in ascending order of magnitude, then:

The median is the middle value of the data set.

The lower quartile (Q1) is the median of the lower half of the data set.

The upper quartile (Q3) is the median of the upper

4 Fit a trend line to the following data by the freehand method:

Year	Production of wheat(in tonnes)	Year	Production of wheat(in tonnes)
1995	20	2000	25
1996	22	2001	23
1997	24	2002	26
1998	21	2003	25
1999	23	2004	24

Answer: Methods of Freehands: It is familiar concept, briefly described for drawing frequency curves. In case of a time series a scatter diagram of the given observations is plotted against time on the horizontal axis and a freehand smooth curve is drawn through the plotted points. The curve is so drawn that most of the points concentrate around the curve, however, smoothness should not be scarified in trying to let the points exactly fall on the curve. It would be better to draw a straight line through the plotted points instead of a curve, if possible. The curve fitted by

5 Let X be a random variable and its probability mass function is

P(x=r) = q^r-1p,r=1,2,3…

Find the m.g.f. of X and hence it’s mean and variance.

Answer: Standard Deviation:-The Standard Deviation is a measure of how spread out numbers are.

Its symbol is σ (the greek letter sigma)

The formula is easy: it is the square root of the Variance. So now you ask, "What is the Variance?"

Variance

6 The diastolic blood pressures of men are distributed as shown in table. Find the standard deviation and variance.

Pressure	78-80	80-82	82-84	84-86	86-88	88-90
No. of Men	3	15	26	23	9	4

Answer: Given values are:

Pressure	78-80	80-82	82-84	84-86	86-88	88-90
No. of Men	3	15	26	23	9	4

Difference = 2

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