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SET 1
- Suppose a person throws a die. If he gets a 5 or 6, he tosses a coin three times and notes the number of heads. If he gets 1, 2, 3 or 4, tosses a coin once and notes whether a head or tail is obtained. If he obtained exactly one head; what is the probability that he threw 5 or 6 with the die.
Ans 1.
The solution to this problem involves Bayes’ theorem, which states that P(A|B) = P(B|A) * P(A) / P(B), where P(A|B) is the probability of event A given event B is true, P(B|A) is the probability of event B given event A is true, P(A) is the probability of event A, and P(B) is the probability of event B.
We want to find the probability that the die
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- A random variable has the following probability distribution
Find (i) the value of ‘’
(iii)
(i)
Solution: Since it is a probability distribution, so, sum of all probabilities must be 1
Since, “a” is a probability, so, it cannot be negative
A = 0.1
(ii) [1.5<x<4.5/x>2]
When 1.5<x<4.5 =2a+3a
=5a
=5(1/10)
=1/2 or 0.5
When x>2 =P(X=3)+ P(X=4)+ P(X=5)+ P(X=6)+ P(X=7)
= 2a+3a+ a2+2a2+7a2
=5a+10a2
=5(1/10)+10(1/10)
- The daily earning of a vendor for a period of 43 days are given below
Daily earning (Rs.) | 118-126 | 127-135 | 136-144 | 145-153 | 154-162 | 163-171 | 172-180 |
No. of days | 3 | 8 | 9 | 12 | 5 | 4 | 2 |
Calculate Standard Deviation and coefficient of variation.
Solution:
Daily earnings | f | x | fx | (x-x̄) | F(x- x̄) | f(x- x̄)2 |
118-126 | 3 | 122 | 366 | -23.86 | -71.58 | 1707.90 |
127-135 | 8 | 131 | 1048 | -14.86 | -118.88 | 1766.56 |
SET-II
- The probability that a bomb dropped from a plane will strike the target is 1/5. If 6 bombs are dropped, find the probability that-
- Exactly two will strike the target.
- Atleast two will strike the target.
Ans 4.
To answer this question, we’ll need to use the binomial probability formula, which is as follows:
P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))
where: P(X=k) is the probability of k successes in n trials, C(n, k) is the number of combinations of n items taken k at a time (also known as “n choose k”), p is the probability of success on a given trial, and (1-p
- Compute the regression equation of Y on X and regression equation of X on Y on the basis of the following information
X | Y | |
Mean | 40 | 45 |
Standard Deviation | 10 | 9 |
The correlation coefficient between X and Y is 0.50.
Also estimate the value of Y for X = 48, using the appropriate regression equation.
Ans 5.
Given that,
- The following data relate to the prices and quantities of 4 commodities in the years 1982 and 1983. Construct the following index numbers of price for the year 1983 by using 1982 as base year.
- Laspeyre’s Index
- Paasche’s Index
iii. Fisher’s Index
Commodity | 1982 | 1983 | ||
Price | Quantity | Price | Quantity | |
A | 5 | 100 | 6 | 150 |
B | 4 | 80 | 5 | 100 |
C | 2.5 | 60 | 5 | 72 |
D | 12 | 30 | 9 | 33 |
Ans 6.
Solution:
Commodit Y
|
1982 | 1983
|
P1q0 | P0q0 | P1q1 | P0q1 | ||
Price(P0) | Quantity(Q0) | Price(P1) | Quantity(Q1) | |||||