5 pair of RVs
5-1: joint pmf
• In a box are three dice. Die 1 is normal; die
2 has no 6 face, but instead two 5 faces;
die 3 has no 5 face, but instead two 6
faces.
• The experiment consists of selecting a die
at random, followed by a toss with that die.
• Let X be the die number that is selected,
and let Y be the face value of that die.
• Find P(X = x, Y = y) for all possible x and y.
5-2: Joint pmf
• A packet switch has two input ports and two
output ports. At a given time slot, a packet
arrives at each input port with probability ½
and is equally likely to be destined to output
port 1 or 2. Let X and Y be the number of
packets destined for output ports 1 and 2,
respectively. Find the joint pmf of (X,Y).
• Three outcomes for each input port can take
the following values:
(i) “n”: no packet arrival;
(ii) “a1”: packet arrival destined for output port 1;
(iii) “a2”: packet arrival destined for output port 2.
5-3: marginal pmf
• We pick a message, whose length N
follows a geometric distribution with
parameter 1-p and SN={0,1,2,…}.
– 𝑃 𝑁 = 𝑘 = 𝑝𝑘 (1 − 𝑝)
• Find the joint pmf and the marginal pmf’s
of Q and R, where Q is the quotient in the
division of N by constant M, and R is the
number of remaining bytes.
5-4: joint cdf
• Joint cdf of (X,Y) is given by
• 𝐹𝑋,𝑌 𝑥, 𝑦 =
•
•
•
•
Find
Find
Find
Find
1 − 𝑒 −𝑥 1 − 𝑒 −𝛽𝑦 𝑖𝑓 𝑥, 𝑦 ≥ 0
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
marginal cdf of X and Y
P[X<1, Y<1]
P[X>x, Y>y]
P[1<X<2, 2<Y<5}
5-5: joint pdf
• Joint pdf is given by:
• Find c.
• Find marginal pdf’s
• Find P[X+Y1]
5-6: independence
• Check whether the following joint pdf is
independent or not
• 𝑓𝑋,𝑌 (𝑥, 𝑦) = 9𝑥 2 𝑦 2
if 0x,y 1
5-7: condi. prob. (discrete)
• The total number of defects X on a chip
is a Poisson random variable with mean .
• Each defect has a probability p of falling
in a specific region R and the location of
each defect is independent of the
locations of other defects.
• Find the pmf of the number of defects Y
that fall in the region R.
5-8: condi. Prob. (conti.)
• X is selected at random from the unit
interval; Y is then selected at random
from the interval (0, X).
• Find the cdf of Y.
5-9: function of RVs
• A system with standby redundancy has a
single key component in operation and a
duplicate of that component in standby
mode.
• When the first component fails, the
second component is put into operation.
• Find the pdf of the lifetime of the standby
system if the components have
independent exponentially distributed
lifetimes with the same mean .