Transcript Vector random variables

6 vector RVs
6-1: probability distribution
• A radio transmitter sends a signal to a
receiver using three paths.
• Let X1, X2, and X3 be the signals that arrive
at the receiver along each path, whose joint
CDF is known.
• Find the probability that maximum signal is
less than or equal to 5.
• Find the probability that X1 is less than of
equal to 5
6-2: joint pmf
6-3: joint pdf
• Let X1 be uniform in [0, 1], X2 be uniform in
[0,X1], and X3 be uniform in [0,X2].
• Note that X3 is also the product of three
uniform random variables.
• Find the joint pdf of X and the marginal
pdf of X3.
6-4: joint characteristic function
• Suppose U and V are independent zeromean, unit-variance Gaussian random
variables
• X = U+V
• Y = 2U+V
• Find the joint characteristic function of X
and Y, and find E[XY].
6-5: sum of geometric RVs
6-6: central limit theorem
• Suppose that orders at a restaurant are iid
random variables with mean  ($8) and
standard deviation  ($2).
• Estimate the probability that the first 100
customers spend a total of more than $840.
• Estimate the probability that the first 100
customers spend a total of between $780
and $820.
• After how many orders can we be 90% sure
that the total spent by all customers is
more than $1000?