IE502: Probabilistic ModelsTutorial

1. A coin is tossed. If it shows heads, you pay Rs.2 (i.e. lose Rs.2). If it showstails, you spin a wheel which gives the amount you win distributed with uniformprobability between Rs.0 and Rs.10. Your gain (or loss) is a random variableX. Find distribution function of X.

2. Consider a random variable X whose mass function is given by :

p(x) =

0.2 if x = −30.1 if x = 0.50.3 if x = 1p/3 if x = 2.9p if x = 40 otherwise

(a) What is p?

(b) Compute P (X2 − 3 > 5).

(c) What is F (1)? What is F (2)? What is F (F (3.1))? (Note: F (·) denotesthe CDF of X).

(d) Plot F (x).

(e) What is P (3X − 3 ≥ 2 | X ≥ 0.75)

(f) Compute E[X].

(g) Compute E[F (x)].

3. Let X and Y have joint density (K is a constant):

f(x, y) =

{K(x+ y)2, for 0 ≤ x, y ≤ 1

0, otherwise

(a) Determine the marginal density of X.

(b) Compute E[X + Y ].

4. Suppose X and Y are i.i.d Exponential random variables with parameter λ.Compute the density of Z = X + Y .

5. You buy a lottery ticket in 50 lotteries, in each of which your chance of winninga prize is 1/100. What is the (approx.) probability that you will win a prize(a) at least once, (b) exactly once, and (c) at least twice?

1