###### Advertisements

###### Advertisements

**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**.

#### 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 | 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.

X | 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 | 0 | 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

x_{i} |
– 2 | – 1 | 0 | 1 | 2 |

p_{i} |
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 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

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 |

- Determine the value of k.
- Find P(X ≤ 4)
- P(2 < X < 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

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

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