STAT311 Final Sample Exam
This is just a sample test
1. One out of 1000 light bulbs manufactured by factory A is defective. What is the probability that
there are more than 2 defective ones in 10,000 light bulbs manufactured by factory A?
1−e−10 −10e−10 −50e−10
2. A deck of 52 cards. Pick one at a time with replacement. If it’s heart, you win $1, if it’s spade, you
lose $1. If it’s club or diamond, you lose 50 cents. On average, how much will you win?
-25cents
3. Coins I and II have probabilities p1and p2of a head respectively. One of the two coins is selected
at random and tossed. Then you put the coin back and randomly select a coin again and toss it.
This process is repeated until the first head occurs. Let Xdenote the number of tosses required
(including the 1st head). Answer questions 3-5.
(a) What is the probability that the 1st head occurs on the 1st toss?
p1+p2
2
(b) What is P(X≥20)?
(1 −p1+p2
2)19
(c) If the 1st head did not occur on the 100th toss, what is the probability that it will occur on the
110th toss?
(2−p1−p2
2)9(p1+p2
2)
4. The joint PMF of Xand Yis given in the table below. Answer questions 6-9.
| Y
|3456
-----------------------------
0 | 0 1/8 1/16 1/8
1 | 0 1/16 3/16 3/16
X 2 | 1/16 0 1/16 0
3 | 0 1/16 0 1/16
(a) P(X= 2, Y > 3) = ?
1
16
(c) P(Y > 3|X= 2) = ?
1
2
(c) Are X,Yindependent random variables? Why?
No, X,Yare not independent. as pX(x)pY(y) is not pX,Y (x, y)
(d) P(2X > Y ) =?
1
8
5. Y1,Y2are independent r.v.’s. Y1∼Poi(λ1), Y2∼Poi(λ2), P(Y1= 0, Y2>4) =?
e−λ1(1 −e−λ2−λ2e−λ2−λ2
2e−λ2
2−λ3
2e−λ2
6)
6. Let Xbe a number picked randomly from (1,2,3). Let Ybe a number picked randomly from
(1,2,3,4). Xand Yare independent. Let Z=X+Y.
(a) What is E(Z)?
4.5
1
