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Assignment 4

This document contains 11 problems related to probability and statistics. Problem 1 asks to find probability functions and construct tables and charts for random variables related to coin tosses. Problem 2 gives a probability mass function and asks to determine values and compute probabilities. Problem 3 gives a probability density function and asks to find values where probabilities are equal or 0.05. The remaining problems involve additional concepts like probability density functions, expected values, variances, and sums of random variables.

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Divyanshu Bose
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185 views3 pages

Assignment 4

This document contains 11 problems related to probability and statistics. Problem 1 asks to find probability functions and construct tables and charts for random variables related to coin tosses. Problem 2 gives a probability mass function and asks to determine values and compute probabilities. Problem 3 gives a probability density function and asks to find values where probabilities are equal or 0.05. The remaining problems involve additional concepts like probability density functions, expected values, variances, and sums of random variables.

Uploaded by

Divyanshu Bose
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Probability and Statistics (MAL-241)

Assignment-4

1. An-experiment consists of three independent tosses of a fair coin. Let


X = The number of heads
Y = The number of head runs,
Z = The length of head runs,

A head run being defined as consecutive occurrence of at least two heads, its length then being the
number of heads occurring together in three tosses of the coin. Find the probability function of (i)
X (ii) Y (iii) Z, (iv) X +Y and (v) XY and construct probability tables and draw their probability
charts.

2. The probability mass function of a random variable X is zero except at the points x  0,1, 2 . At

these points it has the values p(0)  3c 2 ; p(1)  4c  10c2 ; p(2)  5c  1, c  0 .


(i) Determine the value of c. (c=1/3)
(ii) Compute the following probabilities P( X  2) , P(1  X  2) . (1/3, 2/3)

3. A continuous random variable X has a p.d.f. f ( x)  3x 2 ,0  x  1 . Find the value of


a, b such that P( X  a)  P( X  a) , P( X  b)  0.05 .
4. Let X be a continuous, random variable with p.d.f.
ax, 0  x 1
 a, 1 x  2

f ( x)  
ax  3a, 2  x  3
0, elsewhere
Determine the constant a and compute P( X  1.5) . (1/2, 1/2)
5. The time one has to wait for a bus at a downtown bus stop is observed to be random
phenomenon (X) with the following probability density function
0, 0 x
1
 ( x  1), 0  x  1
9
4  1
  x  , 1 x  3 / 2
9  2

f ( x)   4  5 
   x, 3 / 2  x  2
 9 2 
1
 4  x, 2  x  3
9
1
 9 , 3 x 6

Let the events A and B be defined as follows:


A : One waits between 0 to 2 minutes inclusive:
B : One waits between 0 to 3 minutes inclusive.
Draw the graph of probability density function.
Show that P( B | A)  2 / 3, P( A ' B ')  1/ 3

6. A continuous random variable X follows the probability law f ( x)  Ax2 ,0  x  1. Determine


A and find the probability that (i) X lies between 0.2 and 0.5, (ii) X is less than 0.3, (iii)
1/ 4  X  1/ 2 (0.3, 0.117, 0.027, 15/256).
7. Are any of the following probability mass or density functions? Prove your answer in each
case:
(a) f ( x)  x; x  1/16,3 /16,1/ 4,1/ 2

(b) f ( x)  e x ; x  0,   0.

 2 x, 0  x 1

(c) f ( x)  4  2 x, 1 x  2
0,
 elsewhere

8. A continuous random variable X has the probability density function f ( x)  A  Bx;0  x  1.


If the mean (expected value of x) of the distribution is ½. Find A and B.
9. n unbiased dice are thrown. Find the expected values of the sum of numbers of points on them.
(7n/2)
10. If t is any, positive real number, show that the function defined by p( x)  et (1  et ) x1 can
represent a probability function of a random variable X assuming the values 1,2,3,…. Find the
E[ X ],Var[ x]  E[ X 2 ]  ( E[ X ])2 (Ans: et ,(e2t  et ) ).
11. In a lottery m tickets are drawn at a time out of n tickets numbered 1 to n. Find the expectation
of the sum S of the numbers on the tickets drawn. [m(n+1)/2]

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