State whether the following is True or False : If P(X = x) = k[4x] for x = 0, 1, 2, 3, 4 , then F(5) = 14 when F(x) is c.d.f. - Mathematics and Statistics

Advertisements
Advertisements
MCQ
True or False

State whether the following is True or False :

If P(X = x) = `"k"[(4),(x)]` for x = 0, 1, 2, 3, 4 , then F(5) = `(1)/(4)` when F(x) is c.d.f.

Options

  • True

  • False

Advertisements

Solution

False
F (5) = 1.

Concept: Probability Distribution of Discrete Random Variables
  Is there an error in this question or solution?
Chapter 8: Probability Distributions - Miscellaneous Exercise 8 [Page 154]

APPEARS IN

Balbharati Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 8 Probability Distributions
Miscellaneous Exercise 8 | Q 3.01 | Page 154

RELATED QUESTIONS

State if the following is not the probability mass function 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 0.2

State if the following is not the probability mass function of a random variable. Give reasons for your answer

Z 3 2 1 0 −1
P(Z) 0.3 0.2 0.4 0 0.05

State if the following is not the probability mass function of a random variable. Give reasons for your answer.

Y −1 0 1
P(Y) 0.6 0.1 0.2

State if the following is not the probability mass function of a random variable. Give reasons for your answer.

0 -1 -2
P(X) 0.3 0.4 0.3

Find expected value and variance of X for the following p.m.f.

x -2 -1 0 1 2
P(X) 0.2 0.3 0.1 0.15 0.25

Find the mean number of heads in three tosses of a fair coin.


The following is the p.d.f. of r.v. X:

f(x) = `x/8`, for 0 < x < 4 and = 0 otherwise.

Find P (x < 1·5)


It is known that error in measurement of reaction temperature (in 0° c) in a certain experiment is continuous r.v. given by

f (x) = `x^2 /3` , for –1 < x < 2 and = 0 otherwise

 Verify whether f (x) is p.d.f. of r.v. X.


It is known that error in measurement of reaction temperature (in 0° c) in a certain experiment is continuous r.v. given by

f (x) = `x^2/3` , for –1 < x < 2 and = 0 otherwise

Find probability that X is negative


Suppose that X is waiting time in minutes for a bus and its p.d.f. is given by

f (x) = `1/ 5` , for 0 ≤ x ≤ 5 and = 0 otherwise.

Find the probability that waiting time is between 1 and 3


Suppose that X is waiting time in minutes for a bus and its p.d.f. is given by

f (x) = `1/5` , for 0 ≤ x ≤ 5 and = 0 otherwise.

Find the probability that waiting time is more than 4 minutes.


If a r.v. X has p.d.f., 

f (x) = `c /x` , for 1 < x < 3, c > 0, Find c, E(X) and Var (X).


Choose the correct option from the given alternative :

P.d.f. of a.c.r.v X is f (x) = 6x (1 − x), for 0 ≤ x ≤ 1 and = 0, otherwise (elsewhere)

If P (X < a) = P (X > a), then a =


Choose the correct option from the given alternative:

If a d.r.v. X takes values 0, 1, 2, 3, . . . which probability P (X = x) = k (x + 1)·5 −x , where k is a constant, then P (X = 0) =


Choose the correct option from the given alternative:

If p.m.f. of a d.r.v. X is P (X = x) = `((c_(x)^5 ))/2^5` , for x = 0, 1, 2, 3, 4, 5 and = 0, otherwise If a = P (X ≤ 2) and b = P (X ≥ 3), then E (X ) =


Choose the correct option from the given alternative:

If p.m.f. of a d.r.v. X is P (X = x) = `x^2 /(n (n + 1))`, for x = 1, 2, 3, . . ., n and = 0, otherwise then E (X ) =


Choose the correct option from the given alternative :

If p.m.f. of a d.r.v. X is P (x) = `c/ x^3` , for x = 1, 2, 3 and = 0, otherwise (elsewhere) then E (X ) =


Choose the correct option from the given alternative:

If the a d.r.v. X has the following probability distribution :

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

then P (X = −1) =


Solve the following :

The following probability distribution of r.v. X

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


The following is the c.d.f. of r.v. X:

X −3 −2 −1 0 1 2 3 4
F(X) 0.1 0.3 0.5 0.65 0.75 0.85 0.9 1

Find p.m.f. of X.
i. P(–1 ≤ X ≤ 2)
ii. P(X ≤ 3 / X > 0).


The following is the c.d.f. of r.v. X

x -3 -2 -1 0 1 2 3 4
F(X) 0.1 0.3 0.5 0.65 0.75 0.85 0.9

*1

P (–1 ≤ X ≤ 2)


The following is the c.d.f. of r.v. X

x -3 -2 -1 0 1 2 3 4
F(X) 0.1 0.3 0.5 0.65 0.75 0.85 0.9

1

P (X ≤ 3/ X > 0)


Let X be amount of time for which a book is taken out of library by randomly selected student and suppose X has p.d.f

f (x) = 0.5x, for 0 ≤ x ≤ 2 and = 0 otherwise.

Calculate: P(0.5 ≤ x ≤ 1.5)


Find the probability distribution of number of number of tails in three tosses of a coin


Find k if the following function represents the p. d. f. of a r. v. X.

f(x) = `{(kx,  "for"  0 < x < 2),(0,  "otherwise."):}`

Also find `"P"[1/4 < "X" < 1/2]`


Given that X ~ B(n, p), if n = 10 and p = 0.4, find E(X) and Var(X)


X is r.v. with p.d.f. f(x) = `"k"/sqrt(x)`, 0 < x < 4 = 0 otherwise then x E(X) = _______


If F(x) is distribution function of discrete r.v.x with p.m.f. P(x) = `(x - 1)/(3)` for x = 0, 1 2, 3, and P(x) = 0 otherwise then F(4) = _______.


State whether the following is True or False :

x – 2 – 1 1 2
P(X = x) 0.2 0.3 0.15 0.25 0.1

If F(x) is c.d.f. of discrete r.v. X then F(–3) = 0


Solve the following problem :

The probability distribution of a discrete r.v. X is as follows.

X 1 2 3 4 5 6
(X = x) k 2k 3k 4k 5k 6k

Determine the value of k.


Solve the following problem :

Find the expected value and variance of the r. v. X if its probability distribution is as follows.

x 1 2 3
P(X = x) `(1)/(5)` `(2)/(5)` `(2)/(5)`

Solve the following problem :

Let the p. m. f. of the r. v. X be

`"P"(x) = {((3 - x)/(10)", ","for"  x = -1", "0", "1", "2.),(0,"otherwise".):}`
Calculate E(X) and Var(X).


If X denotes the number on the uppermost face of cubic die when it is tossed, then E(X) is ______


If a d.r.v. X takes values 0, 1, 2, 3, … with probability P(X = x) = k(x + 1) × 5–x, where k is a constant, then P(X = 0) = ______


The p.m.f. of a d.r.v. X is P(X = x) = `{{:(((5),(x))/2^5",", "for"  x = 0","  1","  2","  3","  4","  5),(0",", "otherwise"):}` If a = P(X ≤ 2) and b = P(X ≥ 3), then


If a d.r.v. X has the following probability distribution:

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

then P(X = –1) is ______


Find mean for the following probability distribution.

X 0 1 2 3
P(X = x) `1/6` `1/3` `1/3` `1/6`

Find the expected value and variance of r.v. X whose p.m.f. is given below.

X 1 2 3
P(X = x) `1/5` `2/5` `2/5`

The probability distribution of X is as follows:

X 0 1 2 3 4
P(X = x) 0.1 k 2k 2k k

Find k and P[X < 2]


Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as number greater than 4 appears on at least one die.


Choose the correct alternative:

f(x) is c.d.f. of discete r.v. X whose distribution is

xi – 2 – 1 0 1 2
pi 0.2 0.3 0.15 0.25 0.1

then F(– 3) = ______


The values of discrete r.v. are generally obtained by ______


If X is discrete random variable takes the values x1, x2, x3, … xn, then `sum_("i" = 1)^"n" "P"(x_"i")` = ______


E(x) is considered to be ______ of the probability distribution of x.


The probability distribution of a discrete r.v.X is as follows.

x 1 2 3 4 5 6
P(X = x) k 2k 3k 4k 5k 6k

Complete the following activity.

Solution: Since `sum"p"_"i"` = 1

P(X ≤ 4) = `square + square + square + square = square`


The probability distribution of a discrete r.v.X is as follows.

x 1 2 3 4 5 6
P(X = x) k 2k 3k 4k 5k 6k

Complete the following activity.

Solution: Since `sum"p"_"i"` = 1

P(X ≥ 3) = `square - square - square  = square`


Using the following activity, find the expected value and variance of the r.v.X if its probability distribution is as follows.

x 1 2 3
P(X = x) `1/5` `2/5` `2/5`

Solution: µ = E(X) = `sum_("i" = 1)^3 x_"i""p"_"i"`

E(X) = `square + square + square = square`

Var(X) = `"E"("X"^2) - {"E"("X")}^2`

= `sum"X"_"i"^2"P"_"i" - [sum"X"_"i""P"_"i"]^2`

= `square - square`

= `square`


The following function represents the p.d.f of a.r.v. X

f(x) = `{{:((kx;, "for"  0 < x < 2, "then the value of K is ")),((0;,  "otherwise")):}` ______ 


The probability distribution of a discrete r.v. X is as follows:

x 1 2 3 4 5 6
P(X = x) k 2k 3k 4k 5k 6k
  1. Determine the value of k.
  2. Find P(X ≤ 4)
  3. P(2 < X < 4)
  4. P(X ≥ 3)

The probability distribution of X is as follows:

x 0 1 2 3 4
P[X = x] 0.1 k 2k 2k k

Find

  1. k
  2. P[X < 2]
  3. P[X ≥ 3]
  4. P[1 ≤ X < 4]
  5. P(2)

The value of discrete r.v. is generally obtained by counting.


Share
Notifications



      Forgot password?
Use app×