Advertisements
Advertisements
A random variable X has the following probability distribution :
X  0  1  2  3  4  5  6 
P(X)  C  2C  2C  3C  C^{2}  2C^{2}  7C^{2}+C 
Find the value of C and also calculate the mean of this distribution.
Advertisements
Solution
As `sum "P"("X") = 1`
∴ `"C" + 2"C" + 2"C" + 3"C" + "C"^2 + 2"C"^2 + 7"C"^2 + "C" = 1`
⇒ `10"C"^2 + 9"C" 1 = 0`
⇒ `(10"C" 1)("C" + 1)= 0`
∵ `"C" != 1`
so, `"C" = (1)/(10)`.
Also mean = `sum "X""P"("X") = 0 xx "C" + 1 xx 2"C" + 2 xx 2"C" + 3 xx 3"C" + 4 xx "C"^2 + 5 xx 2"C"^2 + 6 xx (7"C"^2 + "C")`
⇒ = `21"C" + 56"C"^2 = 56 xx (1)/(100) + 21 xx (1)/(10) = (266)/(100) or 2.66`.
APPEARS IN
RELATED QUESTIONS
A random variable X has the following probability distribution:
then E(X)=....................
From a lot of 25 bulbs of which 5 are defective a sample of 5 bulbs was drawn at random with replacement. Find the probability that the sample will contain 
(a) exactly 1 defective bulb.
(b) at least 1 defective bulb.
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
Find the probability distribution of number of heads in two tosses of a coin.
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
Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as
(i) number greater than 4
(ii) six appears on at least one die
From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.
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.
The random variable X has probability distribution P(X) of the following form, where k is some number:
`P(X = x) {(k, if x = 0),(2k, if x = 1),(3k, if x = 2),(0, "otherwise"):}`
 Determine the value of 'k'.
 Find P(X < 2), P(X ≥ 2), P(X ≤ 2).
Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.
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).
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]
A random variable X ~ N (0, 1). Find P(X > 0) and P(X < 0).
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.
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.
Which of the following distributions of probabilities of a random variable X are the probability distributions?
(i)
X :  3  2  1  0  −1 
P (X) :  0.3  0.2  0.4  0.1  0.05 
X :  0  1  2 
P (X) :  0.6  0.4  0.2 
(iii)
X :  0  1  2  3  4 
P (X) :  0.1  0.5  0.2  0.1  0.1 
(iv)
X :  0  1  2  3 
P (X) :  0.3  0.2  0.4  0.1 
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.
A random variable X has the following probability distribution:
Values of X :  0  1  2  3  4  5  6  7  8 
P (X) :  a  3a  5a  7a  9a  11a  13a  15a  17a 
Determine:
(i) The value of a
(ii) P (X < 3), P (X ≥ 3), P (0 < X < 5).
The probability distribution function of a random variable X is given by
x_{i} :  0  1  2 
p_{i} :  3c^{3}  4c − 10c^{2}  5c1 
where c > 0 Find: c
Let X be a random variable which assumes values x_{1}, x_{2}, x_{3}, x_{4} such that 2P (X = x_{1}) = 3P(X = x_{2}) = P (X = x_{3}) = 5 P (X = x_{4}). Find the probability distribution of X.
Two cards are drawn from a well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
Four cards are drawn simultaneously from a well shuffled pack of 52 playing cards. Find the probability distribution of the number of aces.
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 without replacement from a well shuffled pack of 52 cards. Find the probability distribution of the number of aces.
Find the probability distribution of the number of white balls drawn in a random draw of 3 balls without replacement, from a bag containing 4 white and 6 red balls
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.
A fair die is tossed twice. If the number appearing on the top is less than 3, it is a success. Find the probability distribution of number of successes.
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 
\[\frac{k}{2}\]

\[\frac{k}{4}\]

\[\frac{k}{8}\]

Determine P(X ≤ 2) and P(X > 2) .
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 x number of colleges. It is given that
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 :
x_{i} :  1  2  3  4 
p_{i}_{ }:  0.4  0.3  0.2  0.1 
Find the mean and standard deviation of each of the following probability distribution :
xi :  0  1  2  3  4  5 
pi : 
\[\frac{1}{6}\]

\[\frac{5}{18}\]

\[\frac{2}{9}\]

\[\frac{1}{6}\]

\[\frac{1}{9}\]

\[\frac{1}{18}\]

A discrete random variable X has the probability distribution given below:
X:  0.5  1  1.5  2 
P(X):  k  k^{2}  2k^{2}  k 
Find the value of k.
Find the mean variance and standard deviation of the following probability distribution
x_{i}_{ }:  a  b 
p_{i} :  p  q 
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 fair die is tossed. Let X denote 1 or 3 according as an odd or an even number appears. 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 .
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.
For what value of k the following distribution is a probability distribution?
X = x_{i} :  0  1  2  3 
P (X = x_{i}) :  2k^{4}  3k^{2} − 5k^{3}  2k − 3k^{2}  3k − 1 
If the probability distribution of a random variable X is given by Write the value of k.
X = x_{i} :  1  2  3  4 
P (X = x_{i}) :  2k  4k  3k  k 
A random variable has the following probability distribution:
X = x_{i} :  1  2  3  4 
P (X = x_{i}) :  k  2k  3k  4k 
Write the value of P (X ≥ 3).
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
Mark the correct alternative in the following question:
For the following probability distribution:
X :  1  2  3  4 
P(X) : 
\[\frac{1}{10}\]

\[\frac{1}{5}\]

\[\frac{3}{10}\]

\[\frac{2}{5}\]

The value of E(X^{2}) is
Mark the correct alternative in the following question:
Let X be a discrete random variable. Then the variance of X is
A die is tossed twice. A 'success' is getting an even number on a toss. Find the variance of number of successes.
Three fair coins are tossed simultaneously. If X denotes the number of heads, find the probability distribution of X.
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
Calculate `"e"_0^circ ,"e"_1^circ , "e"_2^circ` from the following:
Age x  0  1  2 
l_{x}  1000  880  876 
T_{x }      3323 
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).
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.
A fair coin is tossed 12 times. Find the probability of getting exactly 7 heads .
If random variable X has probability distribution function.
f(x) = `c/x`, 1 < x < 3, c > 0, find c, E(x) and Var(X)
If p : It is a day time , q : It is warm
Give the verbal statements for the following symbolic statements :
(a) p ∧ ∼ q (b) p v q (c) p ↔ q
If X ∼ N (4,25), then find P(x ≤ 4)
Find the premium on a property worth ₹12,50,000 at 3% if the property is fully insured.
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 defects on a plywood sheet occur at random with an average of the defect per 50 sq. ft. What Is the probability that such sheet will have
(a) No defects
(b) At least one defect
[Use e^{1} = 0.3678]
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.
An urn contains 5 red and 2 black balls. Two balls are drawn at random. X denotes number of black balls drawn. What are possible values of X?
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.
20 white rats are available for an experiment. Twelve rats are male. Scientist randomly selects 5 rats number of female rats selected on a specific day
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.
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).
A random variable X has the following probability distribution :
x = x  0  1  2  3  7  
P(X=x)  0  k  2k  2k  3k  k^{2}  2k^{2}  7k^{2} + 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  3  4 
P(x)  0.1  0.5  0.2  –0.1  0.3 
Find the probability distribution of the number of successes in two tosses of a die if success is defined as getting a number greater than 4.
A sample of 4 bulbs is drawn at random with replacement from a lot of 30 bulbs which includes 6 defective bulbs. Find the probability distribution of the number of defective bulbs.
A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. If X denotes the age of a randomly selected student, find the probability distribution of X. Find the mean and variance of X.
A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of 2 successes
A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of at least 3 successes
A die is thrown 4 times. If ‘getting an odd number’ is a success, find the probability of at most 2 successes
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 = 0
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
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
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 (i) X = 0, (ii) X ≤ 1, (iii) X > 1, (iv) X ≥ 1.
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
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 (i) no defect, (ii) at least one defect. Use e^{−1} = 0.3678.
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 nonnegative
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 odd.
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 even.
Solve the following problem :
Find the probability of the number of successes in two tosses of a die, where success is defined as six appears in at least one toss.
Solve the following problem :
If a fair coin is tossed 4 times, find the probability that it shows head in the first 2 tosses and tail in last 2 tosses.
Solve the following problem :
The probability that a lamp in the classroom will burn is 0.3. 3 lamps are fitted in the classroom. The classroom is unusable if the number of lamps burning in it is less than 2. Find the probability that the classroom cannot be used on a random occasion.
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 :
In a large school, 80% of the students like mathematics. A visitor asks each of 4 students, selected at random, whether they like mathematics.
Find the probability that the visitor obtains the answer yes from at least 3 students.
Solve the following problem :
It is observed that it rains on 10 days out of 30 days. Find the probability that it rains on at most 2 days of 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
For the random variable X, if V(X) = 4, E(X) = 3, then E(x^{2}) 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
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
Let a pair of dice be thrown and the random variable X be the sum of the numbers that appear on the two dice. Find the mean or expectation of X and variance of X
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 ball drawn, find the probability distribution of 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.
Find the probability distribution of the number of successes in two toves of a die where a success is define as: Six appeared on at least one die.
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  k^{2}  2k^{2}  7k^{2} + 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) = ______.
Box I contains 30 cards numbered 1 to 30 and Box II contains 20 cards numbered 31 to 50. A box is selected at random and a card is drawn from it. The number on the card is found to be a nonprime number. The probability that the card was drawn from Box I is ______.
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 ______.
Find the mean of number randomly selected from 1 to 15.
A random variable X has the following probability distribution:
x  1  2  3  4  5  6  7 
P(x)  k  2k  2k  3k  k^{2}  2k^{2}  7k^{2} + k 
Find:
 k
 P(X < 3)
 P(X > 4)
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 box contains 30 fruits, out of which 10 are rotten. Two fruits are selected at random one by one without replacement from the box. Find the probability distribution of the number of unspoiled fruits. Also find the mean of the probability distribution.