IE 227 RECITATION - 12
19.12.2013
8. 27 Suppose that a petri dish of unit radius, containing nutrients upon which bacteria can
multiply, is smeared with a uniform suspension of bacteria. Subsequently, spots indicating colonies of
bacteria will appear. Suppose that we observe the location of the center of the first spot to appear.
Let Z denotes the distance of the center of the first spot from the center of a petri dish.
a. Determine the CDF of the random variable Z.
b. Use the CDF obtained in part (a) to determine P(a<Z≤b), P(a<Z<b), P(a≤Z<b) and P(a≤Z≤b) for
the specified values of a and b.
i.
a=0.2, b=0.8
ii.
a=0, b=0.8
iii.
a=0.2, b=1.5
Answer: Z is a continuous random variable ranges in the interval [0,1). Therefore, if z<0, we have
F(z)=P(Z<z)=0 and if z≥1, we have F(z)=P(Z<z)=1.
Let 0≤z<1 and the event {Z<z} is that the distance of the center of the first spot from the center of
the petri dish is at most z.
We can take the sample space for this random experiment as Ω={(x,y): x 2+y2<1} So for each event E,
𝑃(𝐸) =
|𝐸| |𝐸|
=
|Ω|
𝜋
…
8. 33 Let X1, …, Xm be independent r.v’s each having the same probability distribution as a random
variable X. Determine the CDF of the following random variables in terms of the CDF of X.
a. 𝑌 = max{𝑋1 , … , 𝑋𝑚 }
b. 𝑍 = min{𝑋1 , … , 𝑋𝑚 }
Answer: 𝑋1 , 𝑋2 , … , 𝑋𝑚 are independent and identical variables such as,
E(X)=E(X1)= … =E(Xm)
and V(X)= V(X1)= … =V(Xm)
a. 𝑌 = max{𝑋1 , … , 𝑋𝑚 }
𝐹𝑌 (𝑦) = 𝑃(𝑌 < 𝑦)
𝑃(𝑌 < 𝑦) = 𝑃(𝑋1 < 𝑦, … , 𝑋𝑚 < 𝑦) = 𝑃(𝑋1 < 𝑦)𝑃( 𝑋2 < 𝑦) … 𝑃(𝑋𝑚 < 𝑦) =
8.69 Define 𝐹: 𝑅 → 𝑅 by 𝐹(𝑥) = 1 − 𝑒 −𝛼𝑥 𝑓𝑜𝑟 𝑥 ≥ 0 𝑎𝑛𝑑 𝐹(𝑥) = 𝑜𝑡𝑤. Show that F is the CDF of a
continuous random variable that is, that F is everywhere continuous and satisfies properties (a) – (d)
of Preposition 8.1 on page 411.
4. Suppose the reaction temperature X (in degrees Celsius) in a certain chemical process has a
uniform distribution with a = -5 and b = 5.
a. Compute P(X < 0).
b. Compute P(-2.5 < X < 2.5)
c. Compute P(-2 ≤ X ≤ 3)
Answer: a.
1
𝐹(𝑥) = {10 − 5 < 𝑥 < 5
0
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
5. Let X denote the amount of time a book on two-hour reserve is actually checked out, and
suppose the cdf is
0
𝑥2
𝐹(𝑥) = {
4
1
Use the cdf to obtain the following:
a. P(X ≤ 1)
b. P(0.5 ≤ X ≤ 1)
c. F’(𝑥) to obtain the density function f(x).
d. E(X)
e.
V(X) and 𝜎𝑥
𝑥<0
0≤𝑥<2
2≤𝑥