Tutorial Sheet No. 3

MAL250(Probability and Stochastic Processes)

1. The percentage of alcohol (100X ) in a certain compound may be considered as a random variable, where X (0 < X < 1) has pdf

fX (x) = 20x3 (1 − x), 0 < x < 1.

Suppose that the selling price of the above compound depends on the alcohol contents. Speciﬁcally, if 1/3 < X < 2/3, the compound sells for c1 dollars/gallon otherwise it sells for c2 dollars/gallon. If the cost is c3 dollars/gallon, ﬁnd the probability distribution of the net proﬁt per gallon. 2. The pdf of a random variable X is given by

fX (x) = 6x(1 − x), 0 < x < 1.

Find the distribution of

(i) Y = X/X+1

2X, −∞ < X < 1/4

1

, 1/4 ≤ X < 3/4

(ii) Y =

2

2

, 3/4 ≤ X < ∞.

3X

3. An angle θ chosen randomly on the interval (0, π/2).What is the probability distribution of X = tanθ? What would be the distribution if θ was to be chosen from (−π/2, π/2) ? 4. A point is chosen randomly on a unit circle. What is the probability distribution of the ordinate of the point so chosen?

5. If X has the standard normal distribution, where X is the position of a randomly moving particle w.r.t. a reference point O on a straight line. What is the distribution of the distance |X | of the point from O?. 6. X has a uniform distribution over the set of integers {-n ,-(n-1),...,-1,0,1,...,(n-1),n}. Find the distribution of (i) |X | (ii) X 2 (iii) 1/1 + |X |.

−

7. If X has N (µ, σ 2 ), ﬁnd the distribution of Y = a + bX , and Z = ( Xσ µ )2 .

8. Consider a nonlinear ampliﬁer whose input X and output Y are related by its transfer characteristic Y

=

=

1

X2,X > 0

1

−|X | 2 , X < 0

Find pdf of Y if X has N (0, 1) distribution.

9. A ﬂuctuating electric current may be considered as a uniformly distributed random variable over the interval (9,11). If this current ﬂows through a 2-ohm register, ﬁnd the pdf of the power P = 2I 2 . 10. The radiant energy EinBhu(hr/f r2 ) is given as the following function of temperature T(in degree Fahrenheit),E = 0.172(T /10)2 .Suppose that the temperature T is considered to be a continuous random variable with pdf fT (t)

200

, 40 ≤ t ≤ 50

t2

= 0, elsewhere.

=

Find the pdf of the radiant energy E.

11. Find the probability density function of an exponential random variable truncated to the left at point a > 0. 12. Find the probability distribution of a binomial random variable X with parameter n, p,truncated to the right at X = r, r > 0.

1

13. Find pdf of a doubly truncated normal N (µ, σ 2 ) random variable, truncated to the left at X = α and to the right at X = β .

14. A ﬁlm supplier produces 10 rolls of a speciﬁcally sensitized ﬁlm each year. If the ﬁlm is not sold within a year it musr be discarded. Past experience indicates that D, the small demand for the ﬁlm, is a Poisson-distributed random variable with parameter 8. If a proﬁt of $7 is made on every roll which is sold, while a loss of $2 is incurred on every roll which must be discarded, compute the expected proﬁt which the supplier may realize on the 10 rolls which he produces.

15. Suppose that D the daily demand for an item,is a random variable with the probability distribution P (D = d

d) = Cd2 , d = 1, 2, 3, 4

!

(a) Evaluate the constant C.

(b) Compute the expected demand.

(c) Suppose that an item is sold for $5.00. A manufacturer produces K item daily. Any item which is not sold at the end of the day must be discarded at a loss of $3.00 (i) ﬁnd the probability distribution of the daily proﬁt as a function of K. (ii)how many items should be manufactured to maximize the expected daily proﬁt?. 16. A certain alloy is formed by combining the melted mixture of two metals. The resulting alloy contains a certain percent of lead, say X , which may be considered as a random variable. Suppose that X has the following pdf:

f (x)

=

6 ∗ 10−6 x(100 − x),

0 ≤ x ≤ 100.

Suppose that P , the net proﬁt realised in selling this alloy per pound, is the...