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Let the p.m.f. of a random variable X be P(x) = 3-x10, for x = −1, 0, 1, 2 = 0, otherwise Then E(x) is - Mathematics and Statistics

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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 ______

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पाठ 2.7: Probability Distributions - MCQ

संबंधित प्रश्‍न

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

X 0 1 2
P (X) 0.4 0.4 0.2

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

X 0 1 2 3 4
P(X) 0.1 0.5 0.2 -0.1 0.3

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 tails in the simultaneous tosses of three coins.


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


Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.


If the probability that a fluorescent light has a useful life of at least 800 hours is 0.9, find the probabilities that among 20 such lights at least 2 will not have a useful life of at least 800 hours. [Given : (0⋅9)19 = 0⋅1348]

 


Three persons A, B and C shoot to hit a target. If A hits the target four times in five trials, B hits it three times in four trials and C hits it two times in three trials, find the probability that:

1) Exactly two persons hit the target.

2) At least two persons hit the target.

3) None hit the target.


A random variable X has the following probability distribution:

Values of X : −2 −1 0 1 2 3
P (X) : 0.1 k 0.2 2k 0.3 k
 

Find the value of k


Let X be a random variable which assumes values x1, x2, x3, x4 such that 2P (X = x1) = 3P(X = x2) = P (X = x3) = 5 P (X = x4). Find the probability distribution of X.                                                                                                                                                                                 


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


Two dice are thrown together and the number appearing on them noted. X denotes the sum of the two numbers. Assuming that all the 36 outcomes are equally likely, what is the probability distribution of X?


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


Let X represent the difference between the number of heads and the number of tails when a coin is tossed 6 times. What are the possible values of X?


The probability distribution of a random variable X is given below:

x 0 1 2 3
P(X) k
\[\frac{k}{2}\]
\[\frac{k}{4}\]
\[\frac{k}{8}\]

 Find P(X ≤ 2) + P(X > 2) .

 

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

xi : 1 2 3 4
pi : 0.4 0.3 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

Determine the mean of the distribution.                


Find the mean variance and standard deviation of the following probability distribution 

xi : a b
pi : p q
where p + q = 1 .

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 coin is tossed four times. Let X denote the number of heads occurring. 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.

 

A fair coin is tossed four times. Let X denote the longest string of heads occurring. Find the probability distribution, mean and variance of X.


Three cards are drawn at random (without replacement) from a well shuffled pack of 52 cards. Find the probability distribution of number of red cards. Hence, find the mean of the distribution .  


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


For what value of k the following distribution is a probability distribution?

X = xi : 0 1 2 3
P (X = xi) : 2k4 3k2 − 5k3 2k − 3k2 3k − 1

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 X is a random-variable with probability distribution as given below:

X = xi : 0 1 2 3
P (X = xi) : k 3 k 3 k k

The value of k and its variance are



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. 


Two fair coins are tossed simultaneously. If X denotes the number of heads, find the probability distribution of X. Also find E(X).


A random variable X has the following probability distribution : 

X = x -2 -1 0 1 2 3
P(x) 0.1 k 0.2 2k 0.3 k

Find the value of k and calculate mean. 


If random variable X has probability distribution function.
f(x) = `c/x`, 1 < x < 3, c > 0, find c, E(x) and Var(X)


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.


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. 


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? 


Find expected value and variance of X, where X is number obtained on uppermost face when a fair die is thrown.


Solve the following:

Identify the random variable as either discrete or continuous in each of the following. Write down the range of it.

A highway safety group is interested in studying the speed (km/hrs) of a car at a check point.


A random variable X has the following probability distribution :

x = x 0 1 2 3       7
P(X=x) 0 k 2k 2k 3k k2 2k2 7k2 + k

Determine (i) k

(ii) P(X> 6)

(iii) P(0<X<3).


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.

x 0 1 2 3 4
P(x) 0.1 0.5 0.2 –0.1 0.3

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

x 0 1 2
P(x) 0.1 0.6 0.3

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

A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of at least 3 successes


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?


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


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.


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 non-negative


Solve the following problem :

The probability that a component will survive a check test is 0.6. Find the probability that exactly 2 of the next 4 components tested survive.


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 0


Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as six appears on at least one die


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

Find the value of k


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).


Find the probability distribution of the maximum of the two scores obtained when a die is thrown twice. Determine also the mean of the distribution.


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 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 random variable x is given as under:
P(X = x) = `{{:("k"x^2,  "for"  x = 1"," 2"," 3),(2"k"x,  "for"  x = 4"," 5"," 6),(0,  "otherwise"):}`
where k is a constant. Calculate E(X)


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

X 2 3 4 5
P(X) `5/"k"` `7/"k"` `9/"k"` `11/"k"`

The value of k is ______.


For the following probability distribution:

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

E(X2) is equal to ______.


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.


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.


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 large chain retailer purchases an electric device from the manufacturer. The manufacturer indicates that the defective rate of the device is 10%. The inspector of the retailer randomly selects 4 items from a shipment. Complete the following activity to find the probability that the inspector finds at most one defective item in the 4 selected items.

Solution:

Here, n = 4

p = probability of defective device = 10% = `10/100 = square`

∴ q = 1 - p = 1 - 0.1 = `square`

X ∼ B(4, 0.1)

 `P(X=x)=""^n"C"_x p^x q^(n-x)= ""^4"C"_x (0.1)^x (0.9)^(4 - x)`

P[At most one defective device] = P[X ≤ 1]

= P[X=0] + P[X=1]

= `square+square`

∴ P[X ≤ 1] = `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.

Kiran plays a game of throwing a fair die 3 times but to quit as and when she gets a six. Kiran gets +1 point for a six and –1 for any other number.

  1. If X denotes the random variable “points earned” then what are the possible values X can take?
  2. Find the probability distribution of this random variable X.
  3. Find the expected value of the points she gets.

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