Tutorial on Probablity
Mathematics Department
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
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