Description

. Given any event E Ã¢Å â€ Ã¢â€žÂ¦, one can always define the indicator random variable IE for this event

as follows:

IE(Ãâ€°) = n

1 if Ãâ€° Ã¢Ë†Ë† E

0 otherwise

Now let A, B, and C denote independent events, with P(A) = P(B) = P(C) = 0.5. Define the

random variable X by X(Ãâ€°) = IA(Ãâ€°) + 2IB(Ãâ€°) Ã¢Ë†â€™ IC(Ãâ€°).

University of California, San Diego

Department of Electrical and Computer Engineering

ECE 109: Problem Set #4

1. Given any event E Ã¢Å â€ Ã¢â€žÂ¦, one can always define the indicator random variable I E for this event

as follows:

n

I E (Ãâ€°) = 1 if Ãâ€° Ã¢Ë†Ë† E

0 otherwise

Now let A, B, and C denote independent events, with P( A) = P( B) = P(C) = 0.5. Define the

random variable X by X (Ãâ€°) = I A (Ãâ€°) + 2I B (Ãâ€°) Ã¢Ë†â€™ IC (Ãâ€°).

(a) What are the values taken on by the random variable X?

(b) Find the cumulative probability distribution function FX (u) and the probability mass function p X (u) of the random variable X. Make sure to specify the values of FX (u) at points

where this function is discontinuous.

2. In this problem, the sample space is Ã¢â€žÂ¦ = {1, 2, 3, 4, 5}. Find a single probability measure on Ã¢â€žÂ¦

(that is, specify the probability of each possible outcome) so that you are able to complete all of

the following tasks. Then complete these tasks.

(a) Define a random variable X on Ã¢â€žÂ¦ that takes the values 1, 2, 3, 4, 5 with probabilities 0.1,

0.1, 0.2, 0.2, 0.4 respectively.

Ã¢Ë†Å¡ Ã¢Ë†Å¡

(b) Define another random variable Y on Ã¢â€žÂ¦ that takes the values 2, 3, Ãâ‚¬ with probabilities

0.2, 0.3, 0.5 respectively.

(c) Consider the random variable Z = XY. Given your answers to parts (a) and (b), what is the

set of possible values of Z? For each zi in this set, what is the probability that Z = zi ?

Hint: There are many different valid answers to parts (a) and (b). Any one of them would suffice.

The answer to part (c) is unique only given the answers to parts (a) and (b).

3. Which of the following are valid cumulative probability distribution functions? For those that are

not valid CDFs, state at least one property of the CDF which is not satisfied. For those that are

valid CDFs, compute the probability P{| X | > 0.5}.

Ã¯Â£Â±

u2

(b) FX (u) =

(

0.5e2u

1 Ã¢Ë†â€™ 0.25eÃ¢Ë†â€™3u

u0

4. The amount of bread (in hundreds of pounds) that a bakery sells in a day is a random variable X

with probability density function

Ã¯Â£Â±

0Ã¢â€°Â¤u0

5e

f X (u) =

0

otherwise

What is the probability that the duration of the conversation

(a) will exceed 5 minutes?

(b) will be less than 6 minutes?

(c) will be between 5 and 6 minutes?

(d) will be less than 6 minutes, given that it was greater than 5 minutes?

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