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महाराष्ट्र राज्य शिक्षण मंडळएचएससी विज्ञान (सामान्य) इयत्ता १२ वी

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) = - Mathematics and Statistics

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प्रश्न

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) =

पर्याय

  • 2x − 3x2

  • 3x − 4x3

  • 3x − 2x3

  • 2x3 − 3x

MCQ
रिकाम्या जागा भरा

उत्तर

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) = 3x − 2x3

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Probability Distribution of Discrete Random Variables
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 7: Probability Distributions - Miscellaneous Exercise 1 [पृष्ठ २४२]

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बालभारती Mathematics and Statistics 2 (Arts and Science) [English] 12 Standard HSC Maharashtra State Board
पाठ 7 Probability Distributions
Miscellaneous Exercise 1 | Q 2 | पृष्ठ २४२

संबंधित प्रश्‍न

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

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

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


Solve the following problem :

A fair coin is tossed 4 times. Let X denote the number of heads obtained. Identify the probability distribution of X and state the formula for p. m. f. of X.


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)


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 ≥ 1.5)


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


Find expected value and variance of X, the number on the uppermost face of a fair die.


70% of the members favour and 30% oppose a proposal in a meeting. The random variable X takes the value 0 if a member opposes the proposal and the value 1 if a member is in favour. Find E(X) and Var(X).


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]`


F(x) is c.d.f. of discrete 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) = _______ .


Choose the correct alternative :

If X ∼ B`(20, 1/10)` then E(X) = _______


Fill in the blank :

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


Fill in the blank :

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


If r.v. X assumes values 1, 2, 3, ……. n with equal probabilities then E(X) = `("n" + 1)/(2)`


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 :

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 1 2 3 ... n
P(X = x) `(1)/"n"` `(1)/"n"` `(1)/"n"` ... `(1)/"n"`

Solve the following problem :

Let X∼B(n,p) If n = 10 and E(X)= 5, find p and Var(X).


Solve the following problem :

Let X∼B(n,p) If E(X) = 5 and Var(X) = 2.5, find n and p.


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 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]


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) = ______


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


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)

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


The p.m.f. of a random variable X is as follows:

P (X = 0) = 5k2, P(X = 1) = 1 – 4k, P(X = 2) = 1 – 2k and P(X = x) = 0 for any other value of X. Find k.


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