11-13. For the given probability density functions f(x), find the cumulative distri-
bution function F(x) =
∫x
−∞
f(x)dx and then plot graphs of both f(x)and F(x).
(a) f(x) =
{
α e−αx, x > 0 and α > 0 constant
0 , x ≤ 0
(b) f(x) =
0 , x ≤− x 0
1
2 x 0 , −x^0 < x < x^0
0 , x ≥x 0 where x 0 > 0 is a constant
(c) f(x) =
√^1
2 π
e−x
(^2) / 2
, −∞ < x < ∞ Leave F(x)in integral form.
(d) f(x) =
{ 1
2 e
x, −∞ < x < 0
1
2 e−x,^0 < x < ∞
11-14. Use factorials to show
(a)
(
n
m
)
=
(
n− 1
m− 1
)
+
(
n− 1
m
)
(b)
(
n
m+ 1
)
=
n−m
m+ 1
(
n
m
)
11-15.
(a) Use a table of areas to find values of tαgiven the area α.
Do for α= 0. 001 , 0. 01 , 0. 025 , 0. 05 , 0. 1
(b) Explain how you would use the table of areas
to calculate the probability P(α < X < β )associated
with a normal distribution (μ= 0, σ = 1).
(c) Use the table of areas to verify (i) P(− 1 < X < 1) ≈ 0. 68 , (ii) P(− 2 < x < 2) ≈ 0. 955 ,
(iii) P(− 3 < X < 3) ≈ 0. 997
11-16. Given an ordinary deck of 52 playing cards.
(a) What is the probability of drawing a black ace?
(b) What is the probability of drawing an ace or a king?
11-17. Given an ordinary deck of 52 playing cards. Let E 1 denote the event of
drawing an ace and E 2 the event of drawing a heart.
(a) Are the events E 1 and E 2 mutually exclusive?
(b) What is the probability of drawing either an ace or a heart or both?
