###### Advertisements

###### Advertisements

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) = ______

#### Options

`7/25`

`16/25`

`18/25`

`19/25`

###### Advertisements

#### Solution

**`16/25`**

#### APPEARS IN

#### 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 |

P(X) |
0.4 | 0.4 | 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 | 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 |

**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 | k^{2} |
2k^{2} |
7k^{2} + k |

**Determine:**

- k
- P(X < 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 |

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)

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

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

P ( 1 < x < 2 )

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

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

P(x > 2)

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)`.

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 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 = 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) =

**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 number of tails in three tosses of a coin

Find the probability distribution of number of heads in four tosses of a coin

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).

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).

**Choose the correct alternative :**

X: is number obtained on upper most face when a fair die….thrown then E(X) = _______.

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

**Fill in the blank :**

If X is discrete random variable takes the value x_{1}, x_{2}, x_{3},…, xn then \[\sum\limits_{i=1}^{n}\text{P}(x_i)\] = _______

**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).

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

Find mean for the following probability distribution.

X |
0 | 1 | 2 | 3 |

P(X = x) |
`1/6` | `1/3` | `1/3` | `1/6` |

**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

x_{i} |
– 2 | – 1 | 0 | 1 | 2 |

p_{i} |
0.2 | 0.3 | 0.15 | 0.25 | 0.1 |

then F(– 3) = ______

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

x |
0 | 1 | 2 |

P(x) |
q^{2} |
2pq | p^{2} |

then Var(x) = ______

If X is discrete random variable takes the values x_{1}, x_{2}, x_{3}, … x_{n}, 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

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`

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 |

- Determine the value of k.
- Find P(X ≤ 4)
- P(2 < X < 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 probability distribution of X is as follows:**

x |
0 | 1 | 2 | 3 | 4 |

P[X = x] |
0.1 | k | 2k | 2k | k |

Find

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

Given below is the probability distribution of a discrete random variable x.

X |
1 | 2 | 3 | 4 | 5 | 6 |

P(X = x) |
K | 0 | 2K | 5K | K | 3K |

Find K and hence find P(2 ≤ x ≤ 3)