ECE-340, Spring 2012
Midterm II Review Questions
1. Suppose that X is a random variable that is uniformly distributed in [0, 1]. Define the
new random variables Y = 2 X -1, Z = X3 and W = u(X - 0.5), where u is the unit-step
function.
a) Determine and plot the pdf of Y.
b) Determine the pdf of Z.
c) Determine and plot the cumulative distribution function of W.
d) Find the mean and variance of W.
e) Classify each random variable in this problem as continuous or discrete, and provide
proper justification.
2. A random variable X has a probability density function fX(x) shown below.
a) Find the value of the constant c.
b) Calculate E[X3].
c) Calculate P{-5 ≤ X ≤ 1}.
d) Calculate P{X2 ≤ 0.2}.
fX(x)
1
x
0
c
3. A random variable X has the cumulative distribution function shown below. It is known
that P{Y=1.5}= 0.55.
FY(y)
a) Correctly label the discontinuity at y=1.5. Justify your
1.0
answer.
c
b) Find the value of c.
c) Plot the pdf of Y.
1.5
0
d) Calculate E[Y].
y
e) Calculate P{Y<0.5}, P{Y ≤ 0.5}, P{Y > 1.5}
4. The number of hits that a website receives between 10AM and 11AM is modeled by a
Poisson random variable, N, with mean 30.
a) Find P{N < 4}.
b) Find the variance of N.
c) Now consider another server that has no association what so ever with the first server.
Assume that the number of hits it receives in the same time period is represented by
another Poisson random variable, M, with mean 20. Find P{M+N <4}. State your
assumptions.
5. In the absence of any voltage source, the voltage V across a resistor is modeled by a zeromean Gaussian random variable with variance V2 = 4kBTR, where T is the temperature of
the resistor in Kelvin (K), kB is the Boltzmann constant (kB=1.3806503 × 10-23 m2 kg s-2
K-1) and R is the resistance in Ohms. This stochastic voltage is simply a result of the
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random motion of all the electrons in the resistor and it is referred to as thermal or
Johnson noise.
a) Write out the pdf of the voltage V.
x
1 t 2 / 2
e
dt ) and V2 .
b) Express P{|V+5| > a} in terms of the  function (  ( x)  

2
c) Suppose that the resistor is connected across the terminals of an ideal (i.e., zero internal
resistance) DC voltage source of value 3 Volts. Write out the pdf of the voltage across the
resistor. [Hint: the source simply adds a DC term to the noise.]
6. Suppose that you shoot a basketball 20 times. The probability of making a successful
throw is 0. 1. Let X represent the number of times you’ve made it.
a) What type of a random variable is X (give its name)? What is P{X=k}
b) What are E[X] and var(X)?
7. Suppose that X and Y are exponentially distributed random variables that are mutually
independent with means  and 4, respectively. Use characteristic functions to determine the
pdf of their sum.
8. Suppose that you play a certain game 20 times independently. The probability of winning
each hand is 0.51, in which case you earn $10; otherwise, you lose $10.
a. What is the probability that you come out even after the twenty plays?
b. If we call your fortune after the 20th hand X, express X in terms of a binomial random
variable. Be specific about the binomial random variable.
c. Calculate E[X] and var(X).
d. Formulate the solution for calculation the probability that you come out a winner
(i.e., X>0) after the 20th hand.
9. Suppose that X is an exponentially distributed random variable with mean .
a. Calculate P{2<X <4};
b. P{X<9|X >5}, how does this compare to P{X<12|X>8}? Comment on your answer.
10. Consider the dart problem discussed in class. Find the probability distribution function of
the distance to the bull’s eye. State any assumption and also specify the probability soace
carefully.
11. Suppose that X is a random variable that is Gaussian with a mean of -1 and a variance of
1 x  1
2. Define the new random variables Y = 0.5 X +1 and Z = g(X), where g ( x)  
.
0 x  1
a. Determine the pdf of Y.
b. Determine and plot the probability mass function of Z.
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ECE-340, Review Questions for Midterm-2, Spring 2012
Problem 1. You are given a probability space (Ω, F, P).
a) Define precisely what is meant by a random variable.
b) Define precisely what is meant by {X ≤ 1} and {X ≤ 1, X > 5}. What is the mathematical
relationship between them and each one of Ω and F?
c) What is mean by the probability distribution function of X. Be precise.
d) What does it mean when we say X is a discrete random variable?
e) What does it mean when we say X is a continuous random variable?
Problem 2. A player shoots a basketball 12 times. Let X represent the number of times the
player makes a basket. Assume that the probability that the player makes a basket in each trial
is p and that trials are mutually independent.
a) What is the probability that the player makes exactly 9 baskets?
b) What is the mean of X?
c) What is the variance of X?
Problem 3. A random variable has a probability density function given by
fX (x) = e−2|x| , x ∈ IR.
a) Is X a continuous or a discrete random variable? Justify your answer.
b) Calculate P{2 ≤ X < 5} and P{X = 0.5}.
c) Calculate and sketch the distribution function of X.
d) Find the probability density function of Y = X 3 .
e) Calculate P{|X − 1| > 5} and P{|X − 2| ≤ 1}.
Problem 4. Let X be a uniformly distributed random variable in the interval [0, 2π].
a) Sketch the probability density function of X and label the graph carefully.
b) Find the mean and variance of Y = 3X − π/2.
c) Sketch the probability density function of Y and label the graph carefully.
Problem 5. Let X be a Gaussian random variable with mean 2 and variance 3.
a) Write out the probability density function of X.
b) Express P{X ≤ 2} in terms of the Q function and evaluate it.
c) Express P{a ≤ X ≤ b} in terms of the Q function.
d) Express P{X ≤ a} in terms of the “erf” function.
e) Suppose that for some z < 2, P{X ≤ z} = 0.1. Calculate P{X ≤ 4 − z}.
Problem 6. A random variable X is exponentially distributed with mean 3.
a) Find the conditional probability distribution function of X given {X > 4}.
b) Find the conditional probability density function of X given {X > 4}.
c) Find the conditional mean of X given {X > 4}.
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