Continuous Random Variables and Problems

CONTINUOUS RANDOM VARIABLES

(i) Definition: Continuous Random Variable

A random variable X is said to be continuous if it takes all possible values between certain limits say from real number 'a' to real number 'b'.

Example: The length of time during which a vacuum tube installed in a circuit functions is a continuous random variable.

Note: If X is a continuous random variable for any x₁ and x₂ P(x₁ ≤ X ≤ x₂) = P(x₁ < X ≤ x₂) = P(x₁ ≤ X < x₂) = P(x₁ < X < x₂)

(ii) Probability density function :

For a continous random variable X, a probability density function is a function such that

(1) f(x) ≥ 0

Note: A probability density function is zero for the values of X which do not occur and it is assumed to be zero wherever it is not specifically defined.

(iii) Cumulative distribution function

The cumulative distribution function of a continuous random variable X is

Note: The probability density function of a continuous random variable can be determined from the cumulative distribution function by differentiating.

(iv) The mean or expected value of a continuous random variable X.

Suppose X is a continuous random variable with probability density function f(x). The mean or expected value of X, denoted as µ or E(X) is

A useful identity is that for any function g,

(v) The variance of a continuous random variable X.

The variance of X, denoted as V(X) or σ², is

Note: The standard deviation of X is σ = 

(vi) FORMULA

Problems based on 

Example 1.5.1

A continuous random variable X has p.d.f. f(x) = k, 0 ≤ x ≤ 1. Determine the constant k. Find P [X ≤ 1/4]

Solution :

Example 1.5.2

Given that the p.d.f of a R.V. X is f(x) = Kx, 0 < x < 1, find K and P(X > 0.5) [A.U. Dec, 96]

Solution :

Example 1.5.3

If f(x) =  is the p.d.f. of a random variable X. Find K. [A.U CBT M/J 2010]

Solution :

Example 1.5.4

A continuous random variable X has probability density function given by f(x) = 3x², 0 ≤ x ≤ 1. Find K such that P(X > K) = 0.5 [A.U. Model Q. Paper] [A.U N/D 2010]

Solution :

Given P(X > K) = 1 - P[X ≤ K] = 1 - 0.5 = 0.5

[Note: P[X ≤ K] = 1 - P [X > K] = 1 - 0.5 = 0.5]

Example 1.5.5

Let X be a continuous R.V with probability density function

Solution :

Example 1.5.6

In a continuous random variable X having the probability density function

 [A.U. A/M 2018 R-13]

(a) Verify  (b) Find P(0 < X ≤ 1)

(c) Find F(x) [cumulative distribution]

Solution :

Example 1.5.7

A continuous random variable X has the density function  Find the value of K and the distribution function. [A.U N/D 2011, N/D 2014] [A.U A/M 2018 R-13]

Solution:

Given f (x) is a p.d.f

Example 1.5.8

The p.d.f. of a continuous R.V. X is f(x) = Ke^-|x|. Find K and the F[x]. [A.U. 2005] [A.U Trichy M/J 2011] [A.U A/M 2010]

Solution :

Given: f(x) is a p.d.f.

Example 1.5.9

A continuous random variable X that can assume any value between x = 2 and x = 5 has a density function given by f(x) = k(1 + x). Find P[X < 4] [A.U M/J 2006, M/J 2007, N/D 2011, N/D 2012] [A.U CBT N/D 2008, CBT Dec. 2009] [A.U A/M 2015, (RP) R13] [A.U A/M 2017 R8]

Solution :

Example 1.5.10

A continuous random variable X has the distribution function find K, probability density function f(x), P[X < 2] [A.U. A/M. 2008]

Solution :

Example 1.5.11

Is the function defined as follows, a density function?

 [A.U N/D 2006]

Solution:

Condition for probability density function is 

Given:

Hence, the given function is density function.

Example 1.5.12

If the density function of a continuous random variable X is given by

(1) Find the value of a.  [A.U A/M 2015 R-8, A.U A/M 2017 R13]

(2) Find the cumulative distribution function of X.

(3) If x₁, x₂ and x₃ are 3 independent observations of X, what is the probability that exactly one of these 3 is greater than 1.5?

Solution:

(1) Since, f (x) is a p.d.f, then

(iv) If 2 ≤ x ≤ 3, then F [x] = 

(v) If x > 3, then F [x] = 

(3) P [X > 1.5] = 

Choosing an X and observing its value can be considered as a trial and X > 1.5 can be considered as a success.

As we choose 3 independent observation of X, n = 3

By Bernoulli's theorem. 

P(exactly one value > 1.5)

Example 1.5.13

Experience has shown that while walking in a certain park, the time X (in mins.), between seeing two people smoking has a density function of the form

 [A.U. N/D 2007]

(1) Calculate the value of ?.

(2) Find the distribution function of X.

(3) What is the probability that Jeff, who has just seen a person smoking, will see another person smoking in 2 to 5 minutes? In atleast 7 minutes ?

Solution:

Given: f(x) = 

Example 1.5.14

The diameter of an electric cable X is a continuous r.v. with pdf f(x) = kx (1-x), 0 ≤ x ≤ 1 [A.U N/D 2017 R-08]

(i) Find the value of k

(ii) c.d.f of X

(iii) the value of a such that P(X < a) = P(X > a)

(iv) 

Solution :

Example 1.5.15

A continuous random variable X that can assume values between x = 2 and x = 5 has dF = [2(1 + x)/27] dx, find P (3 < X < 4)

Solution:

We know that, f(x) = dF/dx = 2 (1 + x)/27, 2 ≤ x ≤ 5

Example 1.5.16

A continuous random variable has the pdf of f(x) = k x⁴; -1 < x < 0. Find the value of k and also 

Solution:

Given: f(x) = kx⁴

(i) To find the value of k

Example 1.5.17

A continuous random variable X has p.d.f f(x) = 3x², 0 ≤ x ≤ 1. Find a and b such (i) P[X = a] = P [X > a] (ii) P[X > b] = 0.05 [A.U M/J 2014 (RP)]

Solution :

Problems based on f(x) = d/dx F[x], mean, variance

Example 1.5.18

The cumulative distribution function (cdf) of a random variable X is  Find the probability density function of X, Mean and variance of X. [AU M/J 2006, AU N/D 2010] [A.U A/M 2015 R8] [A.U A/M 2019 (R17) PS]

Solution:

Given :

 

Example 1.5.19

The sales of a convenience store on a randomly selected day are X thousand dollars, where X is a random variable with a distribution function of the following form: [A.U. N/D 2007]

Suppose that this convenience store's total sales on any given day are less than $ 2000.

(1) Find the value of k

(2) Let A and B be the events that tomorrow the store's total sales are between 500 and 1500 dollars, and over 1000 dollars, respectively. Find P(A) and P(B).

(3) Are A and B independent events?

Solution :

(a) We know that,

Example 1.5.20

If the pdf of a r.v X is given by

 Find P [|X| > 1]

Solution :

P[|X| > 1] = 1 - P[|X| < 1]

Example 1.5.21

If the cdf of a r.v X is given by F(x) =  find the p.d.f of X, also find P [1 ≤ X ≤ 2] [A.U N/D 2010]

Solution :

Example 1.5.22

If f(x) = 

(a) Show that f(x) is a pdf of a continuous random variable X.

(b) Find its distribution function F(x) [A.U CBT A/M 2011]

Solution :

To prove : 

EXERCISES 1.4 and 1.5

1. Define discrete and continuous Random variables with examples.

2. Define probability mass function.

3. Define pdf of a continuous random variables X.

4. Define probability distribution curve.

5. If f (x) is a pdf of a continuous random variables X, then what is the value of 

6. Prove that P (X = c) = 0.

7. For the following density function f (x) =C x² (1-x), 0 < x < 1. Find (i) the constant 'C' (ii) mean. [Ans. (i) C = 12, (ii) mean = 3/5]

8. If the p.d.f of a random variable 'X' is f(x) = y₀ (x - x²) in 0 ≤ x ≤ 1, then find (i) mean (ii) median (iii) mode. [Ans. mean = median = mode = 1/2]

9. Check whether the following are p.d.f. or not.

 [Ans. not a p.d.f.]

10. If f(x) = Ax², 0 ≤ x ≤ 1 is p.d.f. of a random variable 'X', then find

(i) the constant 'A' (ii) P(0.2 < X < 0.5) (iii) P (1/4 < X < 1/2) [Ans. (i) 0.3, (ii) 0.117, (iii) 15/256]

11. Find K, mean and variance if dF = K x² e^-x dx, 0 < x < ∞. [Ans. K = 1/2, mean = 3, variance = 3]

12. Find k so that f(x) given below may be p.d.f  Find the distribution function also. 

13. If the p.d.f of a continuous random variable X is f(x) =  then find the value of c and the distribution function F(x).

[Ans. c = 1/10, F(x) = 

14. Find the value of c and the distribution function F(x), given the p.d.f of a random variables X is given by 

[Ans. c = 2, F(x) = ]

15. If the p.d.f of a random variables 'X' is f(x) = 2x, 0 < x < 1, then find the cdf of X. 

16. If the p.d.f of a random variables X is f (x) = x/2 in 0 ≤ x ≤ 2, then find P(X > 1.5 / X > 1). [Ans. 7/12]

17. The probability mass function of a random variables X is given by  where λ is positive value. Find (i) P(X = 0), (ii) P(X > 2). 

18. Find the cdf whose pdf is given by  and also find P(x ≤ 1/a) and P(1/a < x ≤ 2/a) [Ans ]

19. Given that cdf F (x) = 

(a) Find the pdf of f (x)

(b) Find P(0.5 < x ≤ 0.75)

(c) Find P(x > 0.75) and P(x ≤ 0.5)

[Ans. (a) f (x) =  ; (b) 0.3125; (c) 0.25]

20. The amount of bread (in hundreds of kilos) that a bakery sells in a day is a random variables with density

(i) Find the value of c which makes f(x) a pdf.

(ii) What is the probability that the number of kilos of bread that will be sold in a day is (a) more than 300 kilos (b) between 150 and 450 kilos. [Ans. (i) c = 1/9, (ii) 3/4]

21. A continuous random variables X has a pdf f (x) = 3x², 0 ≤ x ≤ 1. Find a and b such that (i) P(X ≤ a) = P(X > a) and (ii) P(X > b) = 0.05.

[Ans. (i) a =  ; (ii) b = ]

22. Given P(X = x) = , x = 1, 2, ... find P(X ≤ 2), P(X > 3) and c.d.f F(x). [Ans. 0.75, 0.75, F (x) = , x = 1, 2, 3]

23. If the c.d.f of a continuous random variables X is given by F(X) =  then find P (|x| ≤ 1/k). Find the p.d.f f (x). [Ans. P = 1 - 1/e; f(x) = ]

24. If P(x) =  Find (i) P(X = 1 or 2) and (ii)  [Ans. (i) 1/5, (ii) 1/7]

 

25. A continuous random variables X has the density function f(x) given by

26. Find the distribution function of X whose p.d.f. is given by

27. Find P(X = 0) if P(X = 0) = P(X < 0) = P(X > 0) [Ans. 1/3]

28. Two unbiased dice are thrown. Find the expected values of the sum of the numbers on them. [Ans. E (X) = 7]