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महाराष्ट्र राज्य शिक्षण मंडळएचएससी वाणिज्य (इंग्रजी माध्यम) इयत्ता १२ वी

Solve the following problem : Following is the probability distribution of a r.v.X. X – 3 – 2 –1 0 1 2 3 P(X = x) 0.05 0.1 0.15 0.20 0.25 0.15 0.1 Find the probability that X is positive. - Mathematics and Statistics

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प्रश्न

Solve the following problem :

Following is the probability distribution of a r.v.X.

X – 3 – 2 –1 0 1 2 3
P(X = x) 0.05 0.1 0.15 0.20 0.25 0.15 0.1

Find the probability that X is positive.

बेरीज

उत्तर

P(X is positive)

= P(X = 1 or X = 2 or X = 3)

= P(X = 1) + P(X = 2) + P(X = 3)

= 0.25 + 0.15 + 0.10

= 0.50

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पाठ 2.7: Probability Distributions - Short Answers I

संबंधित प्रश्‍न

A random variable X has the following probability distribution:

then E(X)=....................


Probability distribution of X is given by

X = x 1 2 3 4
P(X = x) 0.1 0.3 0.4 0.2

Find P(X ≥ 2) and obtain cumulative distribution function of X


State the following are not the probability distributions of a random variable. Give reasons for your answer.

Z 3 2 1 0 -1
P(Z) 0.3 0.2 0.4 0.1 0.05

Find the probability distribution of number of heads in two tosses of a coin.


Find the probability distribution of number of heads in four tosses of a coin.


A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.


A random variable X has the following probability distribution.

X 0 1 2 3 4 5 6 7
P(X) 0 k 2k 2k 3k k2

2k2

7k2 + k

Determine

(i) k

(ii) P (X < 3)

(iii) P (X > 6)

(iv) P (0 < X < 3)


Two numbers are selected at random (without replacement) from the first six positive integers. Let X denotes the larger of the two numbers obtained. Find E(X).


Assume that the chances of the patient having a heart attack are 40%. It is also assumed that a meditation and yoga course reduce the risk of heart attack by 30% and prescription of certain drug reduces its chances by 25%. At a time a patient can choose any one of the two options with equal probabilities. It is given that after going through one of the two options the patient selected at random suffers a heart attack. Find the probability that the patient followed a course of meditation and yoga?


There are 4 cards numbered 1 to 4, one number on one card. Two cards are drawn at random without replacement. Let X denote the sum of the numbers on the two drawn cards. Find the mean and variance of X.


Let, X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in number of colleges. It is given that

\[P\left( X = x \right) = \begin{cases}kx & , & if x = 0 or 1 \\ 2 kx & , & if x = 2 \\ k\left( 5 - x \right) & , & if x = 3 or 4 \\ 0 & , & if x > 4\end{cases}\]

where k is a positive constant. Find the value of k. Also find the probability that you will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2 colleges.


Two numbers are selected at random (without replacement) from positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the mean and variance of the probability distribution of X.


A random variable X takes the values 0, 1, 2 and 3 such that: 

P (X = 0) = P (X > 0) = P (X < 0); P (X = −3) = P (X = −2) = P (X = −1); P (X = 1) = P (X = 2) = P (X = 3) .  Obtain the probability distribution of X


Find the probability distribution of the number of heads, when three coins are tossed. 


Five defective mangoes are accidently mixed with 15 good ones. Four mangoes are drawn at random from this lot. Find the probability distribution of the number of defective mangoes.


Two cards are drawn successively with replacement from well shuffled pack of 52 cards. Find the probability distribution of the number of aces.


Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of kings.


Find the probability distribution of Y in two throws of two dice, where Y represents the number of times a total of 9 appears.


Three cards are drawn successively with replacement from a well-shuffled deck of 52 cards. A random variable X denotes the number of hearts in the three cards drawn. Determine the probability distribution of X.


An urn contains 4 red and 3 blue balls. Find the probability distribution of the number of blue balls in a random draw of 3 balls with replacement.


From a lot of 10 bulbs, which includes 3 defectives, a sample of 2 bulbs is drawn at random. Find the probability distribution of the number of defective bulbs.


Four balls are to be drawn without replacement from a box containing 8 red and 4 white balls. If X denotes the number of red balls drawn, then find the probability distribution of X.                         


Let, X denote the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in number of colleges. It is given that

\[P\left( X = x \right) = \begin{cases}k\text{ x }  & , & \text{ if } x = 0 \text{ or }  1 \\ 2 \text{ kx }  & , & \text{ if }  x = 2 \\ k\left( 5 - x \right) & , & \text{ if } x = 3 \text{ or } 4 \\ 0 & , & \text{ if } x > 4\end{cases}\]

where k is a positive constant. Find the value of k. Also find the probability that you will get admission in (i) exactly one college (ii) at most 2 colleges (iii) at least 2 colleges.


Find the mean and standard deviation of each of the following probability distribution:

xi :  1 3 4 5
pi:  0.4 0.1 0.2 0.3

 


Find the mean and standard deviation of each of the following probability distribution:

xi : −1 0 1 2 3
pi : 0.3 0.1 0.1 0.3 0.2

Find the mean and standard deviation of each of the following probability distribution :

xi :  -2 -1 0 1 2
pi :  0.1 0.2 0.4 0.2 0.1

A discrete random variable X has the probability distribution given below:

X: 0.5 1 1.5 2
P(X): k k2 2k2 k

Find the value of k.


A discrete random variable X has the probability distribution given below:

X: 0.5 1 1.5 2
P(X): k k2 2k2 k

Determine the mean of the distribution.                


Two cards are drawn simultaneously from a pack of 52 cards. Compute the mean and standard deviation of the number of kings.


Two bad eggs are accidently mixed up with ten good ones. Three eggs are drawn at random with replacement from this lot. Compute the mean for the number of bad eggs drawn.


A pair of fair dice is thrown. Let X be the random variable which denotes the minimum of the two numbers which appear. Find the probability distribution, mean and variance of X.

 

A fair die is tossed. Let X denote twice the number appearing. Find the probability distribution, mean and variance of X.

 

In roulette, Figure, the wheel has 13 numbers 0, 1, 2, ...., 12 marked on equally spaced slots. A player sets Rs 10 on a given number. He receives Rs 100 from the organiser of the game if the ball comes to rest in this slot; otherwise he gets nothing. If X denotes the player's net gain/loss, find E (X).


An urn contains 5 red and 2 black balls. Two balls are randomly drawn, without replacement. Let X represent the number of black balls drawn. What are the possible values of X ? Is X a random variable ? If yes, then find the mean and variance of X.      


If X denotes the number on the upper face of a cubical die when it is thrown, find the mean of X.


If the probability distribution of a random variable X is as given below:

Write the value of P (X ≤ 2).

X = xi : 1 2 3 4
P (X = xi) : c 2c 4c 4c

 

 

If a random variable X has the following probability distribution:

X : 0 1 2 3 4 5 6 7 8
P (X) : a 3a 5a 7a 9a 11a 13a 15a 17a

then the value of a is


A random variable X has the following probability distribution:

X : 1 2 3 4 5 6 7 8
P (X) : 0.15 0.23 0.12 0.10 0.20 0.08 0.07 0.05

For the events E = {X : X is a prime number}, F = {X : X < 4}, the probability P (E ∪ F) is


A random variable has the following probability distribution:

X = xi : 0 1 2 3 4 5 6 7
P (X = xi) : 0 2 p 2 p  3 p  p2 p2 p2 2 p 

The value of p is


Mark the correct alternative in the following question:
Let X be a discrete random variable. Then the variance of X is                

 

 


A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of successes and, hence, find its mean.


Find the probability distribution of the number of doublets in three throws of a pair of dice and find its mean.


From a lot of 15 bulbs which include 5 defective, a sample of 4 bulbs is drawn one by one with replacement. Find the probability distribution of number of defective bulbs. Hence, find the mean of the distribution.     


A die is tossed twice. A 'success' is getting an even number on a toss. Find the variance of number of successes. 


Three cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of spades. Hence, find the mean of the distribtution. 


An urn contains 3 white and 6 red balls. Four balls are drawn one by one with replacement from the urn. Find the probability distribution of the number of red balls drawn. Also find mean and variance of the distribution.


For the following probability density function (p. d. f) of X, find P(X < 1) and P(|x| < 1) 

`f(x) = x^2/18, -3 < x < 3`

            = 0,             otherwise


Demand function x, for a certain commodity is given as x = 200 - 4p where p is the unit price. Find :
(a) elasticity of demand as function of p.
(b) elasticity of demand when p = 10 , interpret your result.


Using the truth table verify that p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r).


If the demand function is D = 150 - p2 - 3p, find marginal revenue, average revenue and elasticity of demand for price p = 3. 


Verify the following function, which can be regarded as p.m.f. for the given values of X : 

X = x -1 0 1
P(x) -0.2 1 0.2

Find mean and standard deviation of the continuous random variable X whose p.d.f. is given by f(x) = 6x(1 - x);= (0);      0 < x < 1(otherwise)


The expenditure Ec of a person with income I is given by E= (0.000035) I2 + (0. 045) I. Find marginal propensity to consume (MPC) and average propensity to consume (APC) when I = 5000.


Alex spends 20% of his income on food items and 12% on conveyance. If for the month of June 2010, he spent ₹900 on conveyance, find his expenditure on food items during the same month. 


The following table gives the age of the husbands and of the wives : 

Age of wives (in years)

Age of husbands (in years)

20-30  30- 40  40- 50  50- 60 
15-25  5 9 3 -
25-35  - 10 25 2
35-45  - 1 12 2
45-55  - - 4 16
55-65  - - - 4

Find the marginal frequency distribution of the age of husbands. 


The p.m.f. of a random variable X is
`"P"(x) = 1/5` , for x = I, 2, 3, 4, 5 
        = 0 , otherwise.
Find E(X).


The probability that a bomb dropped from an aeroplane will strike a target is `1/5`, If four bombs are dropped, find the probability that : 

(a) exactly two will strike the target,
(b) at least one will strike the target. 


Amit and Rohit started a business by investing ₹20,000 each. After 3 months Amit withdrew ₹5,000 and Rohit put in ₹5,000 additionally. How should a profit of ₹12,800 be divided between them at the end of the year? 


A card is drawn at random and replaced four times from a well shuftled pack of 52 cards. Find the probability that -

(a) Two diamond cards are drawn.
(b) At least one diamond card is drawn.


A random variable X has the following probability distribution :

X 0 1 2 3 4 5 6
P(X) C 2C 2C 3C C2 2C2 7C2+C

Find the value of C and also calculate the mean of this distribution.


The p.d.f. of a continuous r.v. X is given by

f (x) = `1/ (2a)` , for 0 < x < 2a and = 0, otherwise. Show that `P [X < a/ 2] = P [X >( 3a)/ 2]` .


The p.d.f. of r.v. of X is given by

f (x) = `k /sqrtx` , for 0 < x < 4 and = 0, otherwise. Determine k .

Determine c.d.f. of X and hence P (X ≤ 2) and P(X ≤ 1).


Determine whether each of the following is a probability distribution. Give reasons for your answer.

x 0 1 2
P(x) 0.4 0.4 0.2

Determine whether each of the following is a probability distribution. Give reasons for your answer.

z 3 2 1 0 -1
P(z) 0.3 0.2 0.4. 0.05 0.05

Determine whether each of the following is a probability distribution. Give reasons for your answer.

y –1 0 1
P(y) 0.6 0.1 0.2

Determine whether each of the following is a probability distribution. Give reasons for your answer.

x 0 1 2
P(x) 0.3 0.4 0.2

A pair of dice is thrown 3 times. If getting a doublet is considered a success, find the probability of two successes


The probability that a bulb produced by a factory will fuse after 200 days of use is 0.2. Let X denote the number of bulbs (out of 5) that fuse after 200 days of use. Find the probability of X ≤ 1


10 balls are marked with digits 0 to 9. If four balls are selected with replacement. What is the probability that none is marked 0?


Find the probability of throwing at most 2 sixes in 6 throws of a single die.


Defects on plywood sheet occur at random with the average of one defect per 50 Sq.ft. Find the probability that such a sheet has no defect


State whether the following is True or False :

If r.v. X assumes the values 1, 2, 3, ……. 9 with equal probabilities, E(x) = 5.


Solve the following problem :

A computer installation has 3 terminals. The probability that any one terminal requires attention during a week is 0.1, independent of other terminals. Find the probabilities that 1 terminal requires attention during a week.


Let the p.m.f. of a random variable X be P(x) = `(3 - x)/10`, for x = −1, 0, 1, 2 = 0, otherwise Then E(x) is ______


A random variable X has the following probability distribution

X 2 3 4
P(x) 0.3 0.4 0.3

Then the variance of this distribution is


Find the probability distribution of the number of doublets in three throws of a pair of dice


Find the mean and variance of the number randomly selected from 1 to 15


A discrete random variable X has the probability distribution given as below:

X 0.5 1 1.5 2
P(X) k k2 2k2 k

Determine the mean of the distribution.


Two biased dice are thrown together. For the first die P(6) = `1/2`, the other scores being equally likely while for the second die, P(1) = `2/5` and the other scores are equally likely. Find the probability distribution of ‘the number of ones seen’.


Two probability distributions of the discrete random variable X and Y are given below.

X 0 1 2 3
P(X) `1/5` `2/5` `1/5` `1/5`

 

Y 0 1 2 3
P(Y) `1/5` `3/10` `2/10` `1/10`

Prove that E(Y2) = 2E(X).


Let X be a discrete random variable whose probability distribution is defined as follows:
P(X = x) = `{{:("k"(x + 1),  "for"  x = 1"," 2"," 3"," 4),(2"k"x,  "for"  x = 5"," 6"," 7),(0,  "Otherwise"):}`
where k is a constant. Calculate the value of k


Let X be a discrete random variable whose probability distribution is defined as follows:
P(X = x) = `{{:("k"(x + 1),  "for"  x = 1"," 2"," 3"," 4),(2"k"x,  "for"  x = 5"," 6"," 7),(0,  "Otherwise"):}`
where k is a constant. Calculate E(X)


Let X be a discrete random variable whose probability distribution is defined as follows:
P(X = x) = `{{:("k"(x + 1),  "for"  x = 1"," 2"," 3"," 4),(2"k"x,  "for"  x = 5"," 6"," 7),(0,  "Otherwise"):}`
where k is a constant. Calculate Standard deviation of X.


The probability distribution of a discrete random variable X is given as under:

X 1 2 4 2A 3A 5A
P(X) `1/2` `1/5` `3/25` `1/10` `1/25` `1/25`

Calculate: The value of A if E(X) = 2.94


The probability distribution of a discrete random variable X is given as under:

X 1 2 4 2A 3A 5A
P(X) `1/2` `1/5` `3/25` `1/10` `1/25` `1/25`

Calculate: Variance of X


A random variable x has to following probability distribution.

X 0 1 2 3 4 5 6 7
P(x) 0 k 2k 2k 3k k2 2k2 7k2 + k

Determine


If the p.m.f of a r. v. X is

P(x) = `c/x^3`, for x = 1, 2, 3

        = 0, otherwise

then E(X) = ______.


Find the mean number of defective items in a sample of two items drawn one-by-one without replacement from an urn containing 6 items, which include 2 defective items. Assume that the items are identical in shape and size.


A person throws two fair dice. He wins ₹ 15 for throwing a doublet (same numbers on the two dice), wins ₹ 12 when the throw results in the sum of 9, and loses ₹ 6 for any other outcome on the throw. Then the expected gain/loss (in ₹) of the person is ______.


Two balls are drawn at random one by one with replacement from an urn containing equal number of red balls and green balls. Find the probability distribution of number of red balls. Also, find the mean of the random variable.


Two numbers are selected from first six even natural numbers at random without replacement. If X denotes the greater of two numbers selected, find the probability distribution of X.


The probability that a bomb will hit the target is 0.8. Complete the following activity to find, the probability that, out of 5 bombs exactly 2 will miss the target.

Solution: Here, n = 5, X =number of bombs that hit the target

p = probability that bomb will hit the target = `square`

∴ q = 1 - p = `square`

Here, `X∼B(5,4/5)`

∴ P(X = x) = `""^"n""C"_x"P"^x"q"^("n" - x) = square`

P[Exactly 2 bombs will miss the target] = P[Exactly 3 bombs will hit the target]

= P(X = 3)

=`""^5"C"_3(4/5)^3(1/5)^2=10(4/5)^3(1/5)^2`

∴ P(X = 3) = `square`


A primary school teacher wants to teach the concept of 'larger number' to the students of Class II. 

To teach this concept, he conducts an activity in his class. He asks the children to select two numbers from a set of numbers given as 2, 3, 4, 5 one after the other without replacement.

All the outcomes of this activity are tabulated in the form of ordered pairs given below:

  2 3 4 5
2 (2, 2) (2, 3) (2, 4)  
3 (3, 2) (3, 3)   (3, 5)
4 (4, 2)   (4, 4) (4, 5)
5   (5, 3) (5, 4) (5, 5)
  1. Complete the table given above.
  2. Find the total number of ordered pairs having one larger number.
  3. Let the random variable X denote the larger of two numbers in the ordered pair.
    Now, complete the probability distribution table for X given below.
    X 3 4 5
    P(X = x)      
  4. Find the value of P(X < 5)
  5. Calculate the expected value of the probability distribution.

Five numbers x1, x2, x3, x4, x5 are randomly selected from the numbers 1, 2, 3, ......., 18 and are arranged in the increasing order such that x1 < x2 < x3 < x4 < x5. What is the probability that x2 = 7 and x4 = 11?


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