STAT 400 Discussion 5 Spring 2015

1. Suppose that the proportion of genetically modified (GMO) corn in a large shipment is 2%. Suppose 50 kernels are randomly and independently selected for testing. a) Find the probability that exactly 2 of these 50 kernels are GMO corn.

b) Use Poisson approximation to find the probability that exactly 2 of these 50 kernels are GMO corn. 2. Suppose a discrete random variable X has the following probability distribution:

f ( 0 ) = 87 , f ( k ) =

k 3

1 , k = 2, 4, 6, 8, … .

( possible values of X are even non-negative integers: 0, 2, 4, 6, 8, … ). Recall Discussion #1 Problem 2 (a): this is a valid probability distribution. a) Find the moment-generating function of X, M X ( t ). For which values of t does it exist? b) Find E ( X ). 3. Suppose a discrete random variable X has the following probability distribution:

f ( 1 ) = ln 3 – 1, f ( k ) = ( )!

3lnk

k, k = 2, 3, 4, … .

( possible values of X are positive integers: 1, 2, 3, 4, … ). Recall Discussion #1 Problem 2 (b): this is a valid probability distribution. a) Find µ X = E ( X ) by finding the sum of the infinite series. b) Find the moment-generating function of X, M X ( t ). c) Use M X ( t ) to find µ X = E ( X ). “Hint”: The answers for (a)

and (c) should be the same.

4. Suppose a discrete random variable X has the following probability distribution:

f ( k ) = k 5

100 , k = 3, 4, 5, 6, … .

Recall Discussion #1 Problem 3: this is a valid probability distribution. a) Find the moment-generating function of X, M X ( t ). For which values of t does it exist? b) Find E ( X ). 5. Let X be a continuous random variable with the probability density function

f ( x ) = 312

2 x , 4 ≤ x ≤ 10, zero otherwise.

a) Find the probability P ( X > 9 ). b) Find the mean of the probability distribution of X. c) Find the median of the probability distribution of X. 6. Suppose a random variable X has the following probability density function: f ( x ) = 1 – x/ 2 , 0 ≤ x ≤ 2, zero elsewhere a) Find the cumulative distribution function F ( x ) = P ( X ≤ x ). b) Find the median of the probability distribution of X. c) Find the probability P( 0.8 ≤ X ≤ 1.8 ). d) Find µX = E ( X ). e) Find σX

2 = Var( X ). f) Find the moment-generating function of X.

1. Suppose that the proportion of genetically modified (GMO) corn in a large shipment is 2%. Suppose 50 kernels are randomly and independently selected for testing. a) Find the probability that exactly 2 of these 50 kernels are GMO corn. Let X = number of GMO kernels in a sample of 50.

Then X has Binomial distribution, n = 50, p = 0.02. P( X = 2 ) = ( ) ( ) 482

0201020250 ..C −⋅⋅ ≈ 0.1858. b) Use Poisson approximation to find the probability that exactly 2 of these 50 kernels are GMO corn. Poisson Approximation to Binomial Distribution:

λ = n ⋅ p = 50 ⋅ 0.02 = 1.0.

P( X = 2 ) = !

201 012 .e. −⋅ ≈ 0.1839.

2. Suppose a discrete random variable X has the following probability distribution:

f ( 0 ) = 87 , f ( k ) =

k 3

1 , k = 2, 4, 6, 8, … .

( possible values of X are even non-negative integers: 0, 2, 4, 6, 8, … ). Recall Discussion #1 Problem 2 (a): this is a valid probability distribution. a) Find the moment-generating function of X, M X ( t ). For which values of t does it exist?

M X ( t ) = E ( e t X ) = ∑∞

=⋅⋅ +

12

2 0

3

1 87

kk

tkt ee = ∑∞

=

+

1

2

9

87

k

kte

=

91

987

2

2

t

t

e

e

−

+ = tt

ee

2

2

987

−+ =

81

9

9 2 −

− te.

Must have

9

2 te < 1 for geometric series to converge. ⇒ t < ln 3.

b) Find E ( X ).

M 'X ( t ) = ( ) ( )( )2 2

2 2 2 2

9

29 2t

tttt

eeeee

−

−−− = ( )2 2

2

9

18t

t

ee

−, t < ln 3.

E ( X ) = M 'X ( 0 ) = 6418 =

329 .

OR

E ( X ) = ∑ ⋅x

xpx

all)( =

8642 3

8

3

6

3

4

3

2870 ++++⋅ + …

91

E ( X ) = 864 3

6

3

4

3

2++ + …

⇒ 98

E ( X ) = 8642 3

2

3

2

3

2

3

2+++ + … =

911

92

− =

41 .

⇒ E ( X ) = 329 .

OR

E ( X ) = ∑ ⋅x

xpx

all)( = ∑

∞

=⋅⋅ +

12 3

12 870

kkk = ∑

∞

=⋅⋅

1 9

1 2k

kk

= ∑∞

=

−⋅⋅⋅

1

1

98

91

82

k

kk = ( ) YE

82 ⋅ ,

where Y has a Geometric distribution with probability of “success” p = 98 .

⇒ E ( X ) = ( ) YE82 ⋅ =

89

82 ⋅ =

329 .

3. Suppose a discrete random variable X has the following probability distribution:

f ( 1 ) = ln 3 – 1, f ( k ) = ( )!

3lnk

k, k = 2, 3, 4, … .

( possible values of X are positive integers: 1, 2, 3, 4, … ). Recall Discussion #1 Problem 2 (b): this is a valid probability distribution.

“Hint”: Recall that ak

ke

ka

0 ! =∑

∞

=.

a) Find µ X = E ( X ) by finding the sum of the infinite series.

E ( X ) = ∑ ⋅x

xpx

all)( = 1 ⋅ ( ln 3 – 1 ) + ( )∑

∞

=⋅

2 !

3ln

k

k

kk

= ln 3 – 1 + ( )( )∑

∞

−=2 !

13ln

k

k

k = ln 3 – 1 + ( ) ( )

( )∑∞

−=

−⋅

2

1

!

13 3 lnln

k

k

k

= ln 3 – 1 + ( ) ( )∑∞

=⋅

1 !

3 3 lnln

k

k

k = ln 3 – 1 + ( ) ( ) 3 1 3 ln

ln −⋅ e

= 3 ln 3 – 1 ≈ 2.2958.

b) Find the moment-generating function of X, M X ( t ).

M X ( t ) = ∑ ⋅x

xt

xp eall

)( = e t ⋅ ( ln 3 – 1 ) + ( )∑∞

=⋅

2 !

3ln

k

kktk

e

= e t ⋅ ( ln 3 – 1 ) + ∑∞

=2 !

3ln

k

kt

k

e = e t ln 3 – e t + 3ln tee – 1 – e t ln 3

= 1 3

−− tete .

c) Use M X ( t ) to find µ X = E ( X ). “Hint”: The answers for (a)

and (c) should be the same.

( ) tt eetet 3 3 M ln

'

X −⋅⋅= , E ( X ) = ( ) 0M ' X = 3 ln 3 – 1.

4. Suppose a discrete random variable X has the following probability distribution:

f ( k ) = k 5

100 , k = 3, 4, 5, 6, … .

Recall Discussion #1 Problem 3: this is a valid probability distribution. a) Find the moment-generating function of X, M X ( t ). For which values of t does it exist?

M X ( t ) = ∑ ⋅x

xt

xp eall

)( = ∑∞

=⋅

3

51100

k

kkt e = ∑

∞

=⋅

3

5100

k

kt e

Geometric series = base1

first term−

=

−

51

5 100

3

t

t

e

e

= t

t

ee

5

4 3

−,

if 5

te < 1 ⇔ t < ln 5.

b) Find E ( X ).

M 'X ( t ) = 2

33

5

45 12

−

−−

−

t

tt tt

e

eeee =

2

43

5

860

−

−

t

t t

e

ee , t < ln 5.

E ( X ) = M 'X ( 0 ) = 1652 =

413 = 3.25.

OR

E ( X ) = ∑ ⋅x

xpx

all)( = 3 × 100

51 3

+ 4 × 100

51 4

+ 5 × 100

51 5

+ …

51

E ( X ) = 3 × 100

51 4

+ 4 × 100

51 5

+ …

⇒ ( 0.8 ) E ( X ) = 200

51 3

+ 100

51 3

+ 100

51 4

+ 100

51 5

+ …

= 1.6 + 1 = 2.6.

⇒ E ( X ) = 8.0

2.6 = 4

13 = 3.25.

5. Let X be a continuous random variable with the probability density function

f ( x ) = 312

2 x , 4 ≤ x ≤ 10, zero otherwise.

a) Find the probability P ( X > 9 ).

P ( X > 9 ) = 936271 xdxx === −

∫ 9367291000

936312

910

310

9

2

≈ 0.2895.

b) Find the mean of the probability distribution of X.

E( X ) = ( )26203 xdxxxdxxfx ==== ∫∫ ⋅⋅

∞

∞− 12489744

1248312

410

410

4

2

≈ 7.8077.

c) Find the median of the probability distribution of X.

F ( x ) = P ( X ≤ x ) = ( )936

64 936312

3

4

3

4

2 −

∞=== ∫∫

−

xydyydyyf x x

x

,

4 ≤ x < 10. F ( x ) = P ( X ≤ x ) = 0, x < 4.

F ( x ) = P ( X ≤ x ) = 1, x ≥ 10.

F ( m ) = 21 .

936643 −m =

21 .

3 m = 642

936+ = 532. m = 3 532 ≈ 8.1028.

6. a) F( x ) = 0 for x ≤ 0,

F( x ) = 4

2 xx − for 0 ≤ x ≤ 2,

F( x ) = 1 for x ≥ 2. b) median = 22− . c) P(0.8 ≤ X ≤ 1.8) = 0.35. d) E(X) = 2/3. e) Var(X) = 2/9. f) Integrating by parts,

M( t ) = 2

2

2

21

ttt e −− , t ≠ 0, M( t ) = 1, t = 0.