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

Question

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

Sum

Solution

The sample space of the experiment is S = {1, 2, 3,…,15}

Let X denotes the selected number.

Then X is a random variable which can take values 1, 2, 3, …, 15.

∴ P(1) = P(2) = P(3) = … = P(15) = `1/15`

E(X) = `sum_("i" = 1)^"n" x_"i""P"_"i"`

= `1 xx 1/15 + 2 xx 1/15 + 3 xx 1/15 + ... + 15 xx 1/15`

= `(1 + 2 + 3 + ... + 15) xx 1/15`

= `((15 xx 16)/2) xx 1/15`

= 8

Var(X) = `(sum_("i" = 1)^"n"x_"i"^2"p"_"i") - (sum_("i" = 1)^"n"x_"i""p"_"i")^2`

= `1^2 xx 1/15 + 2^2 xx 1/15 + 3^2 xx 1/15 + ... + 15^2 xx 1/15 - (8)^2`

= `(1^2 + 2^2 + 3^2 + ... + 15^2) xx 1/15 - (8)^2`

= `((15 xx 16 xx 31)/6) xx 1/15 - (8)^2`

= 82.67 – 64

= 18.67

shaalaa.com

Random Variables and Its Probability Distributions

Is there an error in this question or solution?

Q 5 Q 4 Q 6

Chapter 2.7: Probability Distributions - Short Answers II

APPEARS IN

SCERT Maharashtra Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC

Chapter 2.7 Probability Distributions
Short Answers II | Q 5

Video TutorialsVIEW ALL [3]

view
Video Tutorials For All Subjects
Random Variables and Its Probability Distributions

video tutorial01:49:27
Random Variables and Its Probability Distributions

video tutorial00:14:20
Random Variables and Its Probability Distributions

video tutorial00:44:00

RELATED QUESTIONS

From a lot of 15 bulbs which include 5 defectives, 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.

An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X represents the number of black balls. What are the possible values of X? Is X a random variable?

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

There are 4 cards numbered 1, 3, 5 and 7, 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.

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

(ii)

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 :	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 :	3c³	4c − 10c²	5c-1

where c > 0 Find: c

The probability distribution function of a random variable X is given by

x_i :	0	1	2
p_i :	3c³	4c − 10c²	5c-1

where c > 0 Find: P (X < 2)

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.

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

From a lot containing 25 items, 5 of which are defective, 4 are chosen at random. Let X be the number of defectives found. Obtain the probability distribution of X if the items are chosen without replacement .

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.

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

x_i :	2	3	4
p_i :	0.2	0.5	0.3

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

x_i :	-5	-4	1	2
p_i :	\[\frac{1}{4}\]	\[\frac{1}{8}\]	\[\frac{1}{2}\]	\[\frac{1}{8}\]

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

xi:	0	1	3	5
pi :	0.2	0.5	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.

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

x_i:	a	b
p_i :	p	q

where p + q = 1 .

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

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

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

Write the values of 'a' for which the following distribution of probabilities becomes a probability distribution:

X= x_i:	-2	-1	0	1
P(X= x_i) :	\[\frac{1 - a}{4}\]	\[\frac{1 + 2a}{4}\]	\[\frac{1 - 2a}{4}\]	\[\frac{1 + a}{4}\]

Find the mean of the following probability distribution:

X= x_i:	1	2	3
P(X= x_i) :	\[\frac{1}{4}\]	\[\frac{1}{8}\]	\[\frac{5}{8}\]

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 takes the values 0, 1, 2, 3 and its mean is 1.3. If P (X = 3) = 2 P (X = 1) and P (X = 2) = 0.3, then P (X = 0) is

Mark the correct alternative in the following question:
The probability distribution of a discrete random variable X is given below:

X:	2	3	4	5
P(X):	\[\frac{5}{k}\]	\[\frac{7}{k}\]	\[\frac{9}{k}\]	\[\frac{11}{k}\]

The value of k is .

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

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.

Five bad oranges are accidently mixed with 20 good ones. If four oranges are drawn one by one successively with replacement, then find the probability distribution of number of bad oranges drawn. Hence find the mean and variance of the distribution.

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

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)

John and Mathew started a business with their capitals in the ratio 8 : 5. After 8 months, john added 25% of his earlier capital as further investment. At the same time, Mathew withdrew 20% of bis earlier capital. At the end of the year, they earned ₹ 52000 as profit. How should they divide the profit between them?

A departmental store gives trafnfng to the salesmen in service followed by a test. It is experienced that the performance regarding sales of any salesman is linearly related to the scores secured by him. The following data gives the test scores and sales made by nine (9) salesmen during a fixed period.

Test scores (X)	16	22	28	24	29	25	16	23	24
Sales (Y) (₹ in hundreds)	35	42	57	40	54	51	34	47	45

(a) Obtain the line of regression of Y on X.
(b) Estimate Y when X = 17.

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

If X ∼ N (4,25), then find P(x ≤ 4)

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

Age of wives (in years)	Age of husbands (in years)
Age of wives (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.

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 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 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 (i) X = 0, (ii) X ≤ 1, (iii) X > 1, (iv) X ≥ 1.