Exercises 1.4 and 1.5 (Discrete and Continuous random variable)

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

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 = ½]

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

 (a) f(x) = λe-λx , x ≥ 0 , λ> 0

[ 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 x2e-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

13. If the p.d.f of a continuous random variable X is

 , then find the value of c and the distribution function 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

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

P(x) = α λx / x! , x = 0,1,2,... where λ is positive value. Find (i) P(X = C), (ii) P(X > 2). [Ans. (i) e- λ, (ii) 1- e-λ - e- λ – λ2 e- λ /2]

18. Find the cdf whose pdf is given by f (x) = a2xe-ax and also find P P(x ≤ 1/a )

and P ( 1/a < x  ≤ 2/ a) [Ans. F(x) = 1 - (ax + 1) e-ax, x ≥  p(X  ≤ 1/a ) = 0 0.264; P  P(1/2 <x ≤ 2/a ) = - 0.330 ]

19. Given that cdf 

 (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)

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 = (1/2)1/3 ; (ii) b = (0.95)1/3]

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

23. If the c.d.f of a continuous random variables X is given by

[ Ans. P = 1- 1/e ; f(x) = a/2 e-a:x:]

 [Ans. (i), 1/5 (ii) 1/7 ]

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

(i) f(x) = a / x2 + 1 , -∞ < x < ∞ 

 [Ans. (i) a = 1/π , F(x) = 1/ π (π/2 + tan-1x ), - ∞ < x < ∞ 

(ii) a = 1,  F(x) = 1 – (x + 1 )e-x, x>0 

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]

28. Two unbiased dice are thrown. Find the expected values of the sum of the numbers on them. 

[Ans. E (X) = 7]