Statistical Distribution

Question 1

The random variable X has probability function

P ( X = x ) = Kx, x = 1,2 …, 5

Show that K = 1 / 15

Answer 1

• x = 1, 2, 3, 4 and 5
• K + 2K + 3K + 4K + 5K = 1
• ( Probability is always = 1 )
• 15K = 1
• K = 1 / 15

Question 2

Find

P ( X < 4 )

( The random variable X has probability function

P ( X = x ) = Kx, x = 1,2 …, 5 )

( K = 1 / 15 )

Answer 2

• P ( X < 4 ) = p ( 1 ) + p ( 2 ) + p ( 3 )
• 1 / 15 + 2( 1 / 15 ) + 3( 1 / 15 )
• = 2 / 5

Question 3

The random variable X has probability function

P( X = x ) = { Kx, x = 1, 2, 3
{ K( x + 1 ) x = 4, 5,

where k is a constant.

Find the value of k.

Answer 3

• When x = 1, 2 and 3, Kx is used
• When x = 4 and 5, K( x + 1 ) is used
• K + 2K + 3K + 5K + 6K = 1
• 17K = 1
• K = 1 / 17

Question 4

The random variable X has probability function

P( X = x ) = ( 2x - 1 ) / 36 x = 1, 2, 3, 4, 5, 6.

Construct a table giving the probability distribution of X.

Answer 4

• x 1 2 3 4 5 6
  P ( X = x ) 1/ 36 3 / 36 5 / 36 7 / 36 9 / 36 11 / 36
• ( Values of x were substituted into you the equation )
• ( Fractions were not simplified )

Question 5

Find

P( 2 < X <= 5)

( The random variable X has probability function

P( X = x ) = ( 2x - 1 ) / 36 x = 1, 2, 3, 4, 5, 6. )

( x 1 2 3 4 5 6
P ( X = x ) 1/ 36 3 / 36 5 / 36 7 / 36 9 / 36 11 / 36 )

Answer 5

• 5 / 36 + 7 / 36 + 9 / 36 = 7 / 12

Question 6

The random variable X has probability distribution

x 1 3 5 7 9
P(X = x) 0.2 0.3 0.2 q 0.15

Find the value of q

Answer 6

• 1 - ( 0.2 + 0.3 + 0.2 + 0.15 )

- q = 0.15

Question 7

Find P( 4 < X <= 7 ).

( The random variable X has probability distribution

x 1 3 5 7 9
P(X = x) 0.2 0.3 0.2 q 0.15 )

( q = 0.15 )

Answer 7

• 0.2 + 0.15 = 0.35

Question 8

Tetrahedral dice have four faces.
Two fair tetrahedral dice, one red and one blue, have faces numbered 0, 1, 2, and 3 respectively.
The dice are rolled and the numbers face down on the two dice are recorded.
The random variable R is the score on the red die and the random
variable B is the score on the blue die.

Find P( R = 3 and B = 0 ).

Answer 8

• 1 / 4 X 1 / 4 = 1 / 16

- ( P( R = 3 and B = 0 ) = P( R = 3 n B = 0 ) )

Question 9

The discrete random variable X has probability distribution given by

x -1 0 1 2 3
P( X = x ) 1 / 5 a 1 / 10 a 1 / 5

where a is a constant.

Find the value of a.

Answer 9

• 1 / 5 + a + 1 / 10 + a + 1 / 5 = 1
• 2a = 1 - 1 / 2
• 2a = 1 / 2
• a = 0.25

Question 10

*The random variable Y = 6 - 2X.

Calculate P( X => Y ).

( The discrete random variable X has probability distribution given by

x -1 0 1 2 3
P( X = x ) 1 / 5 a 1 / 10 a 1 / 5 )

( a = 0.25 )

Question 11

The discrete random variable X has the probability distribution

x 1 2 3 4
P( X = x ) k 2k 3k 4k

Two independent observations X1 and X2 are made of X.

Show that P( X1 + X2 = 4 ) = 0.1
( K = 0.1 )

Answer 11

• ( Find x values that make the sum of 4 )
• P( 1 n 3 ), P( 2 n 2 ), P( 3 n 1 )
• P( 1 n 3 ) = 0.1 x 0.3 = 0.03
  ( K = 0.1 ), ( 0.1 x 3( 0.1 ) )
• P( 2 n 2 ) = 0.2 x 0.2 = 0.04
• P( 3 n 1 ) = 0.3 x 0.1 = 0.03
• 0.04 + 0.03 + 0.04 = 0.1

Question 12

Find P( 1.5 < X1 + X2 <= 3.5 )

( The discrete random variable X has the probability distribution

x 1 2 3 4
P( X = x ) k 2k 3k 4k

Two independent observations X1 and X2 are made of X. )

( K = 0.1 )

Answer 12

• ( Choosing values between 2 and 3 due to given range )
• P( 2 ) + P( 3 )
• ( Since we’re looking for X1 + X2 )
• P( 1 n 1 ), makes P( 2 )
• P( 2 n 1 ) and P( 1 n 2 ), makes P( 3 )
• P( 1 n 1 ) = 0.1 x 0.1 = 0.01
• P( 2 n 1 ) = 0.2 x 0.1 = 0.02
• 0.01 + 0.02 + 0.02 = 0.05