The following function represents the p.d.f of a.r.v. X f(x) = forthen the value of K is otherwise{kx;for 0<x<2then the value of K is 0; otherwise ______ - Mathematics and Statistics

Advertisements
Advertisements
MCQ

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")):}` ______ 

Options

  • `3/2`

  • `1/2`

  • 1

  • 0

Advertisements

Solution

`bb(1/2)`

Explanation:

f(x) is a p.d.f of random variable x.

∴ `int_a^b f(x).dx` = 1

⇒ `int_0^2 kx.dx` = 1

⇒ `k[x^2/2]_0^2` = 1

⇒ `k(2^2/2 - 0)` = 1

⇒ 2k = 1

⇒ k = `1/2`.

Concept: Probability Distribution of Discrete Random Variables
  Is there an error in this question or solution?
2021-2022 (March) Set 1

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.

X 0 1 2
P(X) 0.1 0.6 0.3

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.

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

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:

  1. k
  2. P(X < 3)
  3. P( X > 4)

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

Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the standard deviation of 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

 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


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

f(x) = kx(1 – x), for 0 < x < 1 and = 0, otherwise.

Also, find `P(1/4 < x < 1/2) and P(x < 1/2)`.


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.


Choose the correct option from the given alternative:

If the p.d.f of a.c.r.v. X is f (x) = 3 (1 − 2x2 ), for 0 < x < 1 and = 0, otherwise (elsewhere) then the c.d.f of X is F(x) =


Choose the correct option from the given alternative:

If the p.d.f of a.c.r.v. X is f (x) = x`^2/ 18` , for −3 < x < 3 and = 0, otherwise then P (| X | < 1) =


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) = `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) =


Choose the correct option from the given alternative:

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

X -2 -1 0 1 2
P(x) 0.3 0.3 0.1 0.05 0.25

Solve the following :

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

Amount of syrup prescribed by physician.


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)


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

x = x 1 2 3 4 5 6
P[x=x] k 2k 3k 4k 5k 6k

(i) Determine the value of k.

(ii) Find P(X≤4), P(2<X< 4), P(X≥3).


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(x≤1)


Find the probability distribution of number of heads in four 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)


Given that X ~ B(n,p), if n = 25, E(X) = 10, find p and Var (X).


Fill in the blank :

If X is discrete random variable takes the value x1, x2, x3,…, xn then \[\sum\limits_{i=1}^{n}\text{P}(x_i)\] = _______


State whether the following is True or False :

If p.m.f. of discrete r.v. X is

x 0 1 2
P(X = x) q2 2pq p2 

then E(x) = 2p.


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

Find P(X ≤ 4), P(2 < X < 4), P(X ≤ 3).


Solve the following problem :

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

X 0 1 2 3 4 5
P(X = x) `(1)/(32)` `(5)/(32)` `(10)/(32)` `(10)/(32)` `(5)/(32)` `(1)/(32)`

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 the p.m.f. of a d.r.v. X is P(X = x) = `{{:(("c")/x^3",", "for"  x = 1","  2","  3","),(0",", "otherwise"):}` then E(X) = ______


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

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

then k = ______


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`

If p.m.f. of r.v. X is given below.

x 0 1 2
P(x) q2 2pq p2

then Var(x) = ______


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")` = ______


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

k = `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 ≤ 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 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)

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) = _______.


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


Share
Notifications



      Forgot password?
Use app×