Islamic University of Gaza Probability and Stochastic Processes

Faculty of Engineering EELE 3340

Electrical Engineering Department Fall semester 2011/2012

Problem Set #6

Problem(1)

For the following Gaussian R.V,

the peak value is 0.4

(a) Find the expected value and standard deviation.

(b) Sketch the ().

Problem(2)

Find the value of the following probabilities:

a ( > 1.26) b ( < 0.86) c > 1.37 d 1.25 < < 0.37

Problem(3)

Suppose the current in a wire are assumed to follow a normal distribution with a mean of 10 milli-amperes

and a variance of 4 milli-amperes.

(a) What is the probability that a measurement will exceed 13 milli-amperes?

(b) What is the probability that a current is between 9 and 11 milli-amperes?

Problem(4)

Find the value of the following integral:

Problem(5)

A production line manufactures 1000-ohm (R) resistors that have 10 percent tolerance. Let X denotes the

resistance of a resistor. Assuming that X is a normal R.V. with mean 1000 and variance

2500, find the probability that a resistor picked at random will be rejected.

2

2

6 9

2

2

1( )

2

x x

Xf x e

2

2

2

1.2

/2

0

/2

3.5

2 1

2

2

1(a) ( )

2

1(b) ( )

2

1(c) ( )

2

u

U

u

U

u

U

f u e du

f u e du

f u e du

Problem(6)

Consider the function given by

(a) Sketch and show that has the properties of a CDF. (b) Find

i. [ 1

4 ]

ii. [0 < 1

4 ]

iii. [ = 0]

iv. [0 1

4 ]

(c) Specify the type of the R.V X.

Problem(7)

A R.V X is defined by the PDF

(a) Find the value of k.

(b) Find the type of X.

0 0

1 1 0

2 2

11

2

X

x

F (x) x x

x

0 0

1 0 1

2

1

X

x

F (x) x x

k x