Mathematical Biostatistics Boot Camp 1

Quiz

Week 1

1. (1 point) What is P (A B) always equal to?a. 1 P (AC BC)b. 1 P (AC BC)c. 1 P (AC)P (BC)d. P (AC BC)

1.

a.

Solution:

P (A B) = 1 P ((A B)C) = 1 P (AC BC)2. (1 point) Which of the following are always true about P (

ni=1Ei)? (Check all that apply.)It is smaller than or equal to

ni=1 P (Ei).

It is equal to ni=1 P (Ei). It is smaller than maxi P (Ei).It is larger than or equal to mini P (Ei).It is larger than or equal to maxi P (Ei).

It is smaller than mini P (Ei).3. (1 point) Consider inuenza epidemics for two parent heterosexual families. Suppose that the probability is

17% that at least one of the parents has contracted the disease. The probability that the father has contracted

inuenza is 12% while the probability that both the mother and father have contracted the disease is 6%. What

is the probability that the mother has contracted inuenza?

a. 12%

b. 6%

c. 5%

d. 25%

e. 11%

f. 17%

3.

e.

Solution:

Let's have following events:

A : father has the u, and B : mother has the u

We know the following:

P (A B) = 0.17, P (A) = 0.12, and P (A B) = 0.06.From the following formula:

P (A B) = P (A) + P (B) P (A B)we can derive:

P (B) = P (A B) + P (A B) P (A) = 0.17 + 0.06 0.12 = 0.11

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4. (1 point) A random variable, X is uniform, so that it's density is f(x) = 1 for 0 x 1. What is it's 75thpercentile? Express your answer to two decimal places.

a. 0.25

b. 0.50

c. 0.75

d. 0.10

e. 0.90

4.

c.

Solution:

The cumulative distribution function for f(x) is:

F (x) =

x0

f(t) dt =

x0

1 dt = [t]x0 = x 0 = xWe want to nd value of x75 such that F (x75) = 0.75. We know that F (x75) = x75, so x75 = 0.75.

5. (1 point) A Pareto density is

1x2 for 1 < x

6. (1 point) What is the quantile p from the density ex (1 + ex)2?

a. log ((1 p) /p)b. p/ (1 p)c. (1 p) /pd. 1/ (1 + ex)

e. log (p/ (1 p))

6.

e.

Solution:

First, we need to nd cumulative distribution function:

F (x) =

x

et

(1 + et)2dt,

using substitution u(t) = 1 + et and du = et dt we get:

F (x) =

u(x)u()

1u2

du =

1+ex

1u2

du =

[1

u

]1+ex

=1

1 + ex=

exex

ex (ex + 1)=

ex

ex + 1

Now we want to nd a value of xp such that F (xp) = p.

F (xp) = p

exp

exp + 1= p

exp = p (exp + 1)

exp = pexp + p

exp pexp = pexp (1 p) = p

exp =p

1 pxp = log

(p

1 p)

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7. (1 point) Suppose that a density is of the form cxk for some constant c > 1 and 0 < x < 1. What is the valueof c?

a. k + 2

b.

1k

c. 2

d.

1k+1

e. k 1f. k

g. k + 1

7.

g.

Solution: 10

cxkdx = c

[xk+1

k + 1

]10

=c

k + 1

[xk+1

]10=

c

k + 1(1 0) = c

k + 1must be equal to 1, so:

c

k + 1= 1 c = k + 1.

8. (1 point) Suppose that the time in days until hospital discharge for a certain patient population follows a

density f(x) = 12ex/2for x > 0. What is the median discharge time in days?

a. 1.0

b. 1.4

c. 1.8

d. 2.2

e. 2.6

8.

b.

Solution:

First, we need to nd cumulative distribution function F (x):

F (x) =

x0

1

2et/2 dt =

[et/2

]x0=[et/2

]0x= e0 ex/2 = 1 ex/2

Now we need to nd the value of x50 such that F (x50) = 0.5:

F (x50) = 0.5

1 ex50/2 = 0.5ex50/2 = 0.5

x50/2 = log(1

2

)x50/2 = log

(1

2

)x50/2 = log (2)

x50 = 2 log (2) 1.386

R code:

qexp ( 0 . 5 , 1/2)

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9. (1 point) Consider the density given by 2xex2

for x > 0. What is the median?

a. 1.03

b. 0.83

c. 0.79

d. 0.24

e. 0.15

9.

b.

Solution:

First, we need to nd cumulative distribution function F (x):

F (x) =

x0

2tet2

dt,

using substitution u(t) = t2 and du = 2tdt we get:

F (x) = u(x)u(0)

eu du = x20

eu du = [eu]0x2 = e

0 ex2 = 1 ex2

Median is the value of x50 such that F (x50) = 0.5.

F (x50) = 0.5

1 ex250 = 0.5ex

250 = 0.5

x250 = log(1

2

)x250 = log (2)

x50 =log (2) 0.833

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10. (1 point) Suppose h(x) is such that 0 < h(x)