1. Let X be a discrete random variable with the following p.m.f: for a 0, I for aæ = 1, Px (2) = for ar = 2, 0 otherwise. (a) Find Rx, the range of the random variable X, (b) Find P(X > 1.5), (c) Find P(0 < X < 2),

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STAT 221 Prep 1
Random Variables
1. Let X be a discrete random variable with the following p.m.f:
I for x =
0,
for 2=
1,
Px (2) -
for æ = 2,
o otherwise.
(a) Find Rx, the range of the random variable X,
(b) Find P(X > 1.5),
(c) Find P(0 < X < 2),
(d) Find and plot the CDF of X.
2. Let X be a discrete random variable with the following CDF:
(0 for æ < 0,
i for 0<a < 1,
Fx(x) = {} for 1 <a < 2,
1 for 2 <a < 3,
1 for X >> 3.
Find the range and p.m.f. of X.
3. The median of a random variable X is defined as any number m that satisfies
conditions:
1
P(X 2 m) 25
P(X < m) 2 5
and
Note that the median of X is not necessarily unique. Find the median of X if
(a) The p.m.f. of X is given by
0.4
for a = 1,
0.3 for a = 2,
Рx («) —
0.3 for æ = 3,
otherwise.
(b) X is the result of a rolling of a fair die.
4. Let X be a continuous random variable with the following p.d.f.
-4x
се
for æ 20,
fx(2)
otherwise,
where c is a positive constant.
(а) Find c,
-21/31 KD O
Transcribed Image Text:STAT 221 Prep 1 Random Variables 1. Let X be a discrete random variable with the following p.m.f: I for x = 0, for 2= 1, Px (2) - for æ = 2, o otherwise. (a) Find Rx, the range of the random variable X, (b) Find P(X > 1.5), (c) Find P(0 < X < 2), (d) Find and plot the CDF of X. 2. Let X be a discrete random variable with the following CDF: (0 for æ < 0, i for 0<a < 1, Fx(x) = {} for 1 <a < 2, 1 for 2 <a < 3, 1 for X >> 3. Find the range and p.m.f. of X. 3. The median of a random variable X is defined as any number m that satisfies conditions: 1 P(X 2 m) 25 P(X < m) 2 5 and Note that the median of X is not necessarily unique. Find the median of X if (a) The p.m.f. of X is given by 0.4 for a = 1, 0.3 for a = 2, Рx («) — 0.3 for æ = 3, otherwise. (b) X is the result of a rolling of a fair die. 4. Let X be a continuous random variable with the following p.d.f. -4x се for æ 20, fx(2) otherwise, where c is a positive constant. (а) Find c, -21/31 KD O
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