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)=4𝑥+𝑘213𝑥4,0.ifotherwise Find the value of 𝑘.

Q2:

Let 𝑋 be a continuous random variable with the probability density function 𝑓(𝑥)=𝑥82<𝑥<3,1483<𝑥<36,0.ififotherwise Find 𝑃(11𝑥24).

  • A 1 4
  • B7
  • C 1 3 4 8
  • D 1 , 2 8 7 1 6

Q3:

Let 𝑋 be a continuous random variable with probability density function 𝑓(𝑥)=18(6𝑥7)2𝑥3,0.ifotherwise Find 𝑃(2𝑋2.5).

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

Q4:

Let 𝑋 be a continuous random variable with probability density function 𝑓(𝑥)=16(𝑥5)7𝑥9,0.ifotherwise Find 𝑃(𝑋8).

  • A 1 2
  • B 5 1 2
  • C 3 4
  • D0

Q5:

Let 𝑋 be a continuous random variable with probability density function 𝑓(𝑥)=𝑥+𝑘541𝑥7,0.ifotherwise Find 𝑃(𝑋>4).

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

Q6:

Let 𝑋 be a continuous random variable with probability density function 𝑓(𝑥)=124(12𝑥)5𝑥11,0.ifotherwise Find 𝑃(𝑋>7.5).

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

Q7:

Let 𝑋 be a continuous random variable with probability density function 𝑓(𝑥)=𝑎62𝑥30𝑥32,0.ifotherwise Find 𝑃(30.5𝑥31.5).

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

Q8:

Let 𝑋 be a continuous random variable with the probability density function 𝑓(𝑥)=𝑎8𝑥39𝑥41,0.ifotherwise Find the value of 𝑎.

  • A 1 5
  • B 1 1 0
  • C 9 1 0
  • D10

Q9:

Let 𝑋 be a continuous random variable with probability density function 𝑓(𝑥)=7𝑥+4𝑘3𝑥5,0.ifotherwise Find the value of 𝑘.

Q10:

Let 𝑋 be a continuous random variable with probability density function 𝑓(𝑥)=225𝑥+𝑘1𝑥5,0.ifotherwise Find 𝑃(3𝑥4).

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

Q11:

Let 𝑋 be a continuous random variable with the probability density function 𝑓(𝑥)=1639𝑥72,0.ifotherwise Find 𝑃(𝑋>64).

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

Q12:

Let 𝑋 be a continuous random variable with probability density function 𝑓(𝑥)=𝑎𝑥1𝑥5,0.ifotherwise Determine the value of 𝑎.

  • A12
  • B 1 1 2
  • C 1 6
  • D 1 1 1 2

Q13:

Let 𝑋 be a continuous random variable with probability density function 𝑓(𝑥)=128𝑥+𝑘9𝑥11,0.ifotherwise Find the value of 𝑘.

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

Q14:

Let 𝑋 be a continuous random variable with probability density function 𝑓(𝑥)=118(9𝑥)3𝑥9,0.ifotherwise Find 𝑃(4.5𝑋7).

  • A 1 9
  • B1
  • C 6 5 1 4 4
  • D 7 1 6

Q15:

Let 𝑋 be a continuous random variable with probability density function 𝑓(𝑥)=𝑥+2242𝑥6,0.ifotherwise Given 𝑃(𝑎𝑋𝑎+2)=512 and 𝑎[2,6], determine the value of 𝑎.

Q16:

Let 𝑋 be a continuous random variable with probability density function 𝑓(𝑥)=6𝑥+16373𝑥4,0.ifotherwise Given 𝑃(𝑋𝑎)=71148 and 𝑎[3,4], determine the value 𝑎.

Q17:

Let 𝑋 be a continuous random variable with probability density function 𝑓(𝑥)=12(𝑥3)3𝑥5,0.ifotherwise Find 𝑃(𝑋>4.5).

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

Q18:

Let 𝑋 be a continuous random variable with probability density function 𝑓(𝑥)=𝑥456<𝑥<9,1509<𝑥<34,0.ififotherwise Given that 𝑃(13<𝑋<𝑎)=1350, find the value of 𝑎.

Q19:

Let 𝑋 be a continuous random variable with the probability density function 𝑓(𝑥)=7𝑥+2𝑘2𝑥4,0.ifotherwise Find 𝑃(2𝑥2.5).

  • A 7 1 3 6 8
  • B0
  • C 7 1 7 3 6
  • D 2 9 7 3 6 8

Q20:

Let 𝑋 be a continuous random variable with probability density function 𝑓(𝑥)=1403𝑥43,0.ifotherwise Find 𝑃(𝑋=37).

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

Q21:

Let 𝑥 be a continuous random variable with probability density function 𝑓(𝑥)=𝑘5𝑥33,0.ifotherwise Find the value of 𝑘.

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

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(2.98𝑍𝑘)=0.7074.

Q24:

Suppose that 𝑍 is a standard normal random variable. Given that 𝑃(𝑍𝑘)=0.4483, find 𝑘 using the standard normal distribution table.

Q25:

Suppose 𝑍 is a standard normal random variable. Given 𝑃(𝑍𝑘)=0.9922, find the value of 𝑘.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.