Worksheet: Continuous Random Variables

In this worksheet, we will practice describing the probability density function of a continuous random variable and using it to find the probability for some event.

Q1:

Let 𝑋 be a continuous random variable with probability density function 𝑓 ( 𝑥 ) = 4 𝑥 + 𝑘 2 1 3 𝑥 4 , 0 . i f o t h e r w i s e Find the value of 𝑘 .

Q2:

Let 𝑋 be a continuous random variable with the probability density function 𝑓 ( 𝑥 ) = 𝑥 8 2 < 𝑥 < 3 , 1 4 8 3 < 𝑥 < 3 6 , 0 . i f i f o t h e r w i s e Find 𝑃 ( 1 1 𝑥 2 4 ) .

  • A7
  • B 1 , 2 8 7 1 6
  • C 1 4
  • D 1 3 4 8

Q3:

Let be a continuous random variable with probability density function Find .

  • A
  • B0
  • C
  • D

Q4:

Let 𝑋 be a continuous random variable with probability density function 𝑓 ( 𝑥 ) = 1 6 ( 𝑥 5 ) 7 𝑥 9 , 0 . i f o t h e r w i s e Find 𝑃 ( 𝑋 8 ) .

  • A 1 2
  • B 3 4
  • C0
  • D 5 1 2

Q5:

Let 𝑋 be a continuous random variable with probability density function 𝑓 ( 𝑥 ) = 𝑥 + 𝑘 5 4 i f 1 𝑥 7 , 0 o t h e r w i s e . Find 𝑃 ( 𝑋 > 4 ) .

  • A 1 6
  • B 1 1 1 2
  • C 1 0 2 7
  • D 7 1 2

Q6:

Let be a continuous random variable with probability density function Find .

  • A
  • B
  • C
  • D

Q7:

Let 𝑋 be a continuous random variable with probability density function 𝑓 ( 𝑥 ) = 𝑎 6 2 𝑥 3 0 𝑥 3 2 , 0 . i f o t h e r w i s e Find 𝑃 ( 3 0 . 5 𝑥 3 1 . 5 ) .

  • A 1 2 7 4 9 6
  • B 1 2 1 4 9 6
  • C 3 7 5 4 9 6
  • D 1 2

Q8:

Let 𝑋 be a continuous random variable with the probability density function 𝑓 ( 𝑥 ) = 𝑎 8 𝑥 3 9 𝑥 4 1 , 0 . i f o t h e r w i s e Find the value of 𝑎 .

  • A 9 1 0
  • B10
  • C 1 5
  • D 1 1 0

Q9:

Let 𝑋 be a continuous random variable with probability density function 𝑓 ( 𝑥 ) = 7 𝑥 + 4 𝑘 3 𝑥 5 , 0 . i f o t h e r w i s e Find the value of 𝑘 .

Q10:

Let 𝑋 be a continuous random variable with probability density function 𝑓 ( 𝑥 ) = 2 2 5 𝑥 + 𝑘 1 𝑥 5 , 0 . i f o t h e r w i s e Find 𝑃 ( 3 𝑥 4 ) .

  • A 3 3 1 0 0
  • B 1 4
  • C 9 1 0 0
  • D 2 9 1 0 0

Q11:

Let 𝑋 be a continuous random variable with the probability density function 𝑓 ( 𝑥 ) = 1 6 3 9 𝑥 7 2 , 0 . i f o t h e r w i s e Find 𝑃 ( 𝑋 > 6 4 ) .

  • A 1 6 6 3
  • B 4 6 3
  • C 5 5 6 3
  • D 8 6 3

Q12:

Let 𝑋 be a continuous random variable with probability density function 𝑓 ( 𝑥 ) = 𝑎 𝑥 1 𝑥 5 , 0 . i f o t h e r w i s e Determine the value of 𝑎 .

  • A 1 1 1 2
  • B12
  • C 1 6
  • D 1 1 2

Q13:

Let 𝑋 be a continuous random variable with probability density function 𝑓 ( 𝑥 ) = 1 2 8 𝑥 + 𝑘 9 𝑥 1 1 , 0 . i f o t h e r w i s e Find the value of 𝑘 .

  • A4
  • B 1 2 1
  • C 3 4
  • D 1 7

Q14:

Let 𝑋 be a continuous random variable with probability density function 𝑓 ( 𝑥 ) = 1 1 8 ( 9 𝑥 ) 3 𝑥 9 , 0 . i f o t h e r w i s e Find 𝑃 ( 4 . 5 𝑋 7 ) .

  • A 1 9
  • B 7 1 6
  • C1
  • D 6 5 1 4 4

Q15:

Let 𝑋 be a continuous random variable with probability density function 𝑓 ( 𝑥 ) = 𝑥 + 2 2 4 2 𝑥 6 , 0 . i f o t h e r w i s e Given 𝑃 ( 𝑎 𝑋 𝑎 + 2 ) = 5 1 2 and 𝑎 [ 2 , 6 ] , determine the value of 𝑎 .

Q16:

Let 𝑋 be a continuous random variable with probability density function 𝑓 ( 𝑥 ) = 6 𝑥 + 1 6 3 7 3 𝑥 4 , 0 . i f o t h e r w i s e Given 𝑃 ( 𝑋 𝑎 ) = 7 1 1 4 8 and 𝑎 [ 3 , 4 ] , determine the value 𝑎 .

Q17:

Let 𝑋 be a continuous random variable with probability density function 𝑓 ( 𝑥 ) = 1 2 ( 𝑥 3 ) i f 3 𝑥 5 , 0 o t h e r w i s e . Find 𝑃 ( 𝑋 > 4 . 5 ) .

  • A 3 4
  • B 3 3 1 6
  • C 1 5 1 6
  • D 7 1 6

Q18:

Let 𝑋 be a continuous random variable with probability density function 𝑓 ( 𝑥 ) = 𝑥 4 5 i f 6 < 𝑥 < 9 , 1 5 0 i f 9 < 𝑥 < 3 4 , 0 o t h e r w i s e . Given that 𝑃 ( 1 3 < 𝑋 < 𝑎 ) = 1 3 5 0 , find the value of 𝑎 .

Q19:

Let 𝑋 be a continuous random variable with the probability density function 𝑓 ( 𝑥 ) = 7 𝑥 + 2 𝑘 2 𝑥 4 , 0 . i f o t h e r w i s e Find 𝑃 ( 2 𝑥 2 . 5 ) .

  • A0
  • B 7 1 7 3 6
  • C 2 9 7 3 6 8
  • D 7 1 3 6 8

Q20:

Let 𝑋 be a continuous random variable with probability density function 𝑓 ( 𝑥 ) = 1 4 0 3 𝑥 4 3 , 0 . i f o t h e r w i s e Find 𝑃 ( 𝑋 = 3 7 ) .

  • A 3 1 0
  • B 3 2 0
  • C 1 7 2 0
  • D 1 4 0

Q21:

Let 𝑥 be a continuous random variable with probability density function 𝑓 ( 𝑥 ) = 𝑘 i f 5 𝑥 3 3 , 0 o t h e r w i s e . Find the value of 𝑘 .

  • A 1 5 6
  • B 2 7 2 8
  • C 1 2 8
  • D 1 2 8

Q22:

Calculate the probability density function at 𝑥 = 1 for an exponential distribution with 𝜆 = 1 .

Q23:

Let 𝑍 be a standard normal random variable. Determine 𝑘 , given that 𝑃 ( 2 . 9 8 𝑍 𝑘 ) = 0 . 7 0 7 4 .

Q24:

Suppose that 𝑍 is a standard normal random variable. Given that 𝑃 ( 𝑍 𝑘 ) = 0 . 4 4 8 3 , find 𝑘 using the standard normal distribution table.

Q25:

Suppose 𝑍 is a standard normal random variable. Given 𝑃 ( 𝑍 𝑘 ) = 0 . 9 9 2 2 , find the value of 𝑘 .

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