Quiz 5B for MATH 105 SECTION 205
March 20, 2015
Family Name
Given Name
Student Number
1. Let f (x) =
ae−2x , if x ≥ 1,
for some constant a such that f (x) is a PDF for some continuous random
0,
otherwise
variable X.
(a) (0.5 points) Find a.
(a)
(b) (0.5 points) Find the cumulative distribution function F (x) of X.
(b)
(c) (0.5 points) Compute Pr(X ≤ 2).
(c)
(d) (0.5 points) Compute the mean of X.
(d)
(e) (0.5 points) Compute the variance of X.
(e)
(f) (0.5 points) Compute the standard deviation of X.
(f)
Z
1
dx
.
+ x2
0
(a) (0.5 points) Find the explicit formula of an .
2. For a sequence
{an }∞
n=1
such that an =
√
n2
(a)
(b) (0.5 points) Find a100 .
(b)
(c) (0.5 points) If {an }∞
n=1 has a limit, find this limit.
(c)
3. For a sequence {an }∞
k=1 such that a1 = 1 and an+1 = 4an for all n ≥ 1.
(a) (0.5 points) Compute a2 , a3 , a4 , a5 .
(a)
(b) (0.5 points) Find the explicit formula of an .
(b)
√
cos( n) 3 tan−1 (n)
4. (a) (0.5 points) Compute lim 2 √
+
.
n→∞
n3 + 3
n
(a)
n3 + 2n
23n−1
−1
+ n−10 .
(b) (0.5 points) Compute lim tan
n→∞
n2 + 2n + 1
9
(b)
(c) (0.5 points) Evaluate
∞
X
k=5
1
.
(3k + 1)(3k + 4)
(c)
(d) (0.5 points) Evaluate
∞
X
k=2
ln
k+1
.
k
(d)
(e) (0.5 points) Evaluate
∞
X
k=8
(f) (0.5 points) Is the series
3 · 42n−5
.
74n+5
∞
X
k=100
√
(e)
k+1
convergent or divergent?
k
(f)
(g) (0.5 points) Is the series
∞
X
4
convergent or divergent?
k ln2 k
k=9
(g)
ax2 + b, if 0 ≤ x ≤ 1,
for some constants a and b such that f (x) is a probability density
0,
otherwise
function for some continuous random variable X.
5. Let f (x) =
(a) (1 point) Find conditions for a and b.
(b) (1 point) Compute a and b such that E(X) = 1 and σ(X) = 2.
Your Score:
/11