###### Advertisements

###### Advertisements

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)

###### Advertisements

#### Solution

(X ≤ 3) ∩ (X > 0)

= {-3, -2, -1, 0, 1, 2, 3} ∩ {1, 2, 3, 4}

= {1, 2, 3}

∴ P[(X ≤ 3) ∩ ( X > 0)]

= P(X = 1)+ P(X = 2)+ P(X = 3)

∴ P[(X ≤ 3) / ( X > 0)]

= `(P[(X ≤ 3) ∩ ( X > 0)])/(P(X > 0))`

= `(P(X = 1) + P(X = 2) + P(X=3))/(P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4))`

= `(0.1 + 0.1 + 0.05)/(0.1 + 0.1 + 0.05 + 0.1)`

=`0.25/0.35`

= `5/7`

#### 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.

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 |

Y |
−1 | 0 | 1 |

P(Y) |
0.6 | 0.1 | 0.2 |

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

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

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 represent p.d.f. of r.v. X

f (x) = kx, for 0 < x < 2 and = 0 otherwise, Also find P `(1/ 4 < x < 3 /2)`.

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 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 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) = `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 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 :**

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.

**Solve the following :**

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

The person on the high protein diet is interested gain of weight in a week.

**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 |

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 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

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

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)

**Choose the correct alternative :**

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

**Choose the correct alternative :**

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

**Solve the following problem :**

The p.m.f. of a r.v.X is given by

`P(X = x) = {(((5),(x)) 1/2^5", ", x = 0", "1", "2", "3", "4", "5.),(0,"otherwise"):}`

Show that P(X ≤ 2) = 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 | 0 | 1 |

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

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) = ______

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) = ______

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

**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) = ______

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

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`

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")):}` ______

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) = _______.

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)