Question

Q6/

Let X be a discrete random variable defined by the following probability function

x | 2 | 3 | 7 | 9 |

f(x) | 0.15 | 0.25 | 0.35 | 0.25 |

Give P(4≤ X < 8)

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Q7/

Let X be a discrete random variable defined by the following probability function

x | 2 | 3 | 7 | 9 |

f(x) | 0.15 | 0.25 | 0.35 | 0.25 |

Let F(x) be the CDF of X. Give F(7.5)

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Q8/

Let X be a discrete random variable defined by the following probability function :

x | 2 | 6 | 9 | 13 |

f(x) | 0.25 | 0.15 | 0.25 | 0.35 |

Give E(X)

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Q9/ Let X be a discrete random variable defined by the following probability function :

x | 1 | 5 | 7 | 11 |

f(x) | 0.15 | 0.25 | 0.35 | 0.25 |

Give Var(X).

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Q10 / (Let X be a discrete random variable with variance Var(X)=3. Find Var(4X +7

Answer #1

Suppose that the probability mass function for a discrete random
variable X is given by p(x) = c x, x = 1, 2, ... , 9. Find the
value of the cdf (cumulative distribution function) F(x) for 7 ≤ x
< 8.

Let x be a discrete random variable with the following
probability distribution
x: -1 , 0 , 1, 2
P(x) 0.3 , 0.2 , 0.15 , 0.35
Find the mean and the standard deviation of x

1. Let X be a discrete random variable with the probability mass
function P(x) = kx2 for x = 2, 3, 4, 6.
(a) Find the appropriate value of k.
(b) Find P(3), F(3), P(4.2), and F(4.2).
(c) Sketch the graphs of the pmf P(x) and of the cdf F(x).
(d) Find the mean µ and the variance σ 2 of X. [Note: For a
random variable, by definition its mean is the same as its
expectation, µ = E(X).]

Let the probability function of the random variable X be f(x) =
{ x⁄45 if x = 1, 2, 3, ⋯ ⋯ ,9 ; 0 otherwise}
Find E(X) and Var(X)

Let X be a discrete random variable with probability mass
function (pmf) P (X = k) = C *ln(k) for k = e; e^2 ; e^3 ; e^4 ,
and C > 0 is a constant.
(a) Find C.
(b) Find E(ln X).
(c) Find Var(ln X).

Problem 3. Let x be a discrete random variable with the
probability distribution given in the following table:
x = 50 100 150 200 250 300 350
p(x) = 0.05 0.10 0.25 0.15 0.15 0.20 0.10
(i) Find µ, σ 2 , and σ.
(ii) Construct a probability histogram for p(x).
(iii) What is the probability that x will fall in the interval
[µ − σ, µ + σ]?

Let X be a discrete random variable with the range RX = {1, 2,
3, 4}. Let PX(1) = 0.25, PX(2) = 0.125, PX(3) = 0.125.
a) Compute PX(4).
b) Find the CDF of X.
c) Compute the probability that X is greater than 1 but less
than or equal to 3.

Let X be a discrete random variable that takes on the values −1,
0, and 1. If E (X) = 1/2 and Var(X) = 7/16, what is the probability
mass function of X?

Let X be a continuous random variable with probability density
function (pdf) ?(?) = ??^3, 0 < ? < 2.
(a) Find the constant c.
(b) Find the cumulative distribution function (CDF) of X.
(c) Find P(X < 0.5), and P(X > 1.0).
(d) Find E(X), Var(X) and E(X5 ).

Compute the mean and standard deviation of the random variable
with the given discrete probability distribution.
x
Px
−3
0.23
−2
0.15
7
0.25
9
0.27
10
0.1

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