Mathematics · Textbook solutions

Probability Distributions

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 99 questions

7.1-7.2 Random Variables and Their Types

1 q

Illustrations

Worked · 1
  1. 7.1 ForExample.1
    Three seeds are sown in order to find how many of them germinate. Every seed will either germinate or will not germinate. Use the letter YY when a seed germinates and the letter NN when a seed does not germinate. Write the sample space of this experiment. Let XX denote the number of times the letter YY appears in a possible outcome. Find the value of XX for every outcome, the range of XX, and the four events [X=0][X = 0], [X=1][X = 1], [X=2][X = 2], [X=3][X = 3].

7.3 Probability Distribution of a Discrete Random Variable

12 q

Illustrations

Worked · 6
  1. 7.3 ForExample.1
    Consider the experiment of throwing two dice and noting the numbers on the upper-most faces of the two dice. Let XX denote the sum of the two numbers in any single throw. Obtain the probability distribution of XX.
  2. 7.3.1 ForExample.1
    Consider the coin-tossing experiment where the random variable XX is defined as the number of tosses required to get a head. Let the probability of getting head be tt and that of not getting head be 1−t1 - t. Write the probability mass function of XX and verify the result.
  3. 7.3.1 ForExample.2
    Consider the experiment of tossing a coin 4 times and defining the random variable XX as the number of heads in 4 tosses. Write the probability distribution of XX in tabular form and state the p.m.f. of XX.
  4. 7.3.2 ForExample.1
    Consider the experiment of tossing 4 coins and counting the number of heads. Let XX denote the number of heads obtained. Form the table showing the p.m.f. f(x)=P[X=x]f(x) = P[X = x] and the c.d.f. F(x)=P[X≤x]F(x) = P[X \leq x] of XX.
  5. 7.3.2 ForExample.2
    Consider the experiment of tossing a coin till a head is obtained. Let the random variable XX denote the number of tosses required for the first head. Show the p.m.f. f(x)f(x) and the c.d.f. F(x)F(x) of XX in tabular form.
  6. 7.3.2 ForExample.3
    Consider the simple experiment of tossing a coin twice. The sample space of this experiment is S={HH, HT, TH, TT}S = \{HH,\ HT,\ TH,\ TT\}. Let XX denote the number of heads obtained in two tosses, let YY denote the number of heads minus the number of tails in two tosses, and let Z=Number of headsNumber of tails+1Z = \dfrac{\text{Number of heads}}{\text{Number of tails} + 1}. Obtain the value of each of XX, YY and ZZ for every outcome of the experiment.

Solved Examples

Worked · 6
  1. 7.3.2 SolvedEx.1
    Two persons A and B play a game of tossing a coin thrice. If the result of a toss is head, A gets ₹ 2 from B. If the result of a toss is tail, B gets ₹ 1.5 from A. Let XX denote the amount gained or lost by A. Show that XX is a discrete random variable and show how it can be defined as a function on the sample space of the experiment.
  2. 7.3.2 SolvedEx.2
    A bag contains 1 red and 2 green balls. One ball is drawn from the bag at random, its colour is noted, and then ball is put back in the bag. One more ball is drawn from the bag at random and its colour is also noted. Let XX denote the number of red balls drawn from the bag as described above. Derive the probability distribution of XX.
  3. 7.3.2 SolvedEx.3
    Two cards are randomly drawn, with replacement, from a well shuffled deck of 52 playing cards. Find the probability distribution of the number of aces drawn.
  4. 7.3.2 SolvedEx.4
    A fair die is thrown. Let XX denote the number of factors of the number on the upper face. Find the probability distribution of XX.
  5. 7.3.2 SolvedEx.5
    Find the probability distribution of the number of doublets in three throws of a pair of dice.
  6. 7.3.2 SolvedEx.6
    The probability distribution of XX is as follows :
    xx01234
    P[X=x]P[X = x]0.1kk2k2k2k2kkk
    Find (i) kk, (ii) P[X<2]P[X < 2], (iii) P[X≥3]P[X \geq 3], (iv) P[1≤X<4]P[1 \leq X < 4], (v) F(2)F(2).

7.3.3 Expected Value and Variance

4 q

Solved Examples

Worked · 4
  1. 7.3.3 SolvedEx.1
    Three coins are tossed simultaneously, XX is the number of heads. Find expected value and variance of XX.
  2. 7.3.3 SolvedEx.2
    Let a pair of dice be thrown and the random variable XX be the sum of the numbers that appear on the two dice. Find the mean or expectation of XX and variance of XX.
  3. 7.3.3 SolvedEx.3
    Find the mean and variance of the number randomly selected from 1 to 15.
  4. 7.3.3 SolvedEx.4
    Two cards are drawn simultaneously (or successively without replacement) from a well shuffled pack of 52 cards. Find the mean, variance and standard deviation of the number of kings drawn.

Exercise 7.1

25 q
  1. Ex 7.1 Q.1
    Let XX represent the difference between number of heads and number of tails obtained when a coin is tossed 6 times. What are possible values of XX ?
  2. Ex 7.1 Q.2
    An urn contains 5 red and 2 black balls. Two balls are drawn at random. XX denotes number of black balls drawn. What are possible values of XX ?
  3. Ex 7.1 Q.3 (i)
    State which of the following are not the probability mass function of a random variable. Give reasons for your answer.
    XX012
    P(X)P(X)0.40.40.2
  4. Ex 7.1 Q.3 (ii)
    State which of the following are not the probability mass function of a random variable. Give reasons for your answer.
    XX01234
    P(X)P(X)0.10.50.2−0.1-0.10.2
  5. Ex 7.1 Q.3 (iii)
    State which of the following are not the probability mass function of a random variable. Give reasons for your answer.
    XX012
    P(X)P(X)0.10.60.3
  6. Ex 7.1 Q.3 (iv)
    State which of the following are not the probability mass function of a random variable. Give reasons for your answer.
    ZZ3210−1-1
    P(Z)P(Z)0.30.20.400.05
  7. Ex 7.1 Q.3 (v)
    State which of the following are not the probability mass function of a random variable. Give reasons for your answer.
    YY−1-101
    P(Y)P(Y)0.60.10.2
  8. Ex 7.1 Q.3 (vi)
    State which of the following are not the probability mass function of a random variable. Give reasons for your answer.
    XX0−1-1−2-2
    P(X)P(X)0.30.40.3
  9. Ex 7.1 Q.4 (i)
    Find the probability distribution of number of heads in two tosses of a coin.
  10. Ex 7.1 Q.4 (ii)
    Find the probability distribution of number of tails in the simultaneous tosses of three coins.
  11. Ex 7.1 Q.4 (iii)
    Find the probability distribution of number of heads in four tosses of a coin.
  12. Ex 7.1 Q.5
    Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as number greater than 4 appears on at least one die.
  13. Ex 7.1 Q.6
    From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement. Find the probability distribution of the number of defective bulbs.
  14. Ex 7.1 Q.7
    A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tails.
  15. Ex 7.1 Q.8 (i)
    A random variable XX has the following probability distribution :
    XX01234567
    P(X)P(X)0kk2k2k2k2k3k3kk2k^22k22k^27k2+k7k^2 + k
    Determine kk.
  16. Ex 7.1 Q.8 (ii)
    A random variable XX has the following probability distribution :
    XX01234567
    P(X)P(X)0kk2k2k2k2k3k3kk2k^22k22k^27k2+k7k^2 + k
    Determine P(X<3)P(X < 3).
  17. Ex 7.1 Q.8 (iii)
    A random variable XX has the following probability distribution :
    XX01234567
    P(X)P(X)0kk2k2k2k2k3k3kk2k^22k22k^27k2+k7k^2 + k
    Determine P(X>4)P(X > 4).
  18. Ex 7.1 Q.9
    Find expected value and variance of XX for the following p.m.f.
    XX−2-2−1-1012
    P(X)P(X)0.20.30.10.150.25
  19. Ex 7.1 Q.10
    Find expected value and variance of XX, where XX is number obtained on uppermost face when a fair die is thrown.
  20. Ex 7.1 Q.11
    Find the mean number of heads in three tosses of a fair coin.
  21. Ex 7.1 Q.12
    Two dice are thrown simultaneously. If XX denotes the number of sixes, find the expectation of XX.
  22. Ex 7.1 Q.13
    Two numbers are selected at random (without replacement) from the first six positive integers. Let XX denote the larger of the two numbers obtained. Find E(X)E(X).
  23. Ex 7.1 Q.14
    Let XX denote the sum of the numbers obtained when two fair dice are rolled. Find the standard deviation of XX.
  24. Ex 7.1 Q.15
    A class has 15 students whose ages are 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age XX of the selected student is recorded. What is the probability distribution of the random variable XX ? Find mean, variance and standard deviation of XX.
  25. Ex 7.1 Q.16
    In a meeting, 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take X=0X = 0 if he opposed, and X=1X = 1 if he is in favour. Find E(X)E(X) and Var(X)Var(X).

7.4 Continuous Random Variables

9 q

Solved Examples

Worked · 9
  1. 7.4 SolvedEx.1
    Let XX be a continuous random variable whose probability density function is f(x)=3x2f(x) = 3x^2, for 0<x<10 < x < 1. Note that f(x)f(x) is not P[X=x]P[X = x]. For example, f(0.9)=3(0.9)2=2.43>1f(0.9) = 3(0.9)^2 = 2.43 > 1, which is clearly not a probability. In the continuous case, f(x)f(x) is the height of the curve at X=xX = x, so that the total area under the curve is 1. Here it is areas under the curve that define the probabilities. Verify that f(x)f(x) is a proper probability density function. Also, for real numbers cc and dd such that 0≤c<d≤10 \leq c < d \leq 1, find P[c<X<d]P[c < X < d]. Hence answer the following. (a) What is the probability that XX falls between 12\dfrac{1}{2} and 1? That is, what is P[12<X<1]P\left[\dfrac{1}{2} < X < 1\right]? (b) What is P(X=12)P\left(X = \dfrac{1}{2}\right)?
  2. 7.4 SolvedEx.2
    Let XX be a continuous random variable whose probability density function is f(x)=x34f(x) = \dfrac{x^3}{4} for an interval 0<x<c0 < x < c. What is the value of the constant cc that makes f(x)f(x) a valid probability density function?
  3. 7.4 SolvedEx.3
    Let's return to the example in which XX has the following probability density function : f(x)=3x2f(x) = 3x^2 for 0<x<10 < x < 1. What is the cumulative distribution function F(x)F(x)?
  4. 7.4 SolvedEx.4
    Let's return to the example in which XX has the following probability density function : f(x)=x34f(x) = \dfrac{x^3}{4} for 0<x<40 < x < 4. What is the cumulative distribution function XX?
  5. 7.4 SolvedEx.5
    Suppose the p.d.f. of a continuous random variable XX is defined as: f(x)={x+1,for −1<x<01−x,for 0≤x<1f(x) = \begin{cases} x + 1, & \text{for } -1 < x < 0 \\ 1 - x, & \text{for } 0 \leq x < 1 \end{cases} Find the c.d.f. F(x)F(x).
  6. 7.4 SolvedEx.6
    Verify if the following functions are p.d.f. of a continuous r.v. XX. (i) f(x)=e−xf(x) = e^{-x}, for 0<x<∞0 < x < \infty and =0= 0, otherwise. (ii) f(x)=x2f(x) = \dfrac{x}{2}, for −2<x<2-2 < x < 2 and =0= 0, otherwise.
  7. 7.4 SolvedEx.7
    Find kk if the following function is the p.d.f. of r.v. XX. f(x)=kx2(1−x)f(x) = kx^2(1 - x), for 0<x<10 < x < 1 and =0= 0, otherwise.
  8. 7.4 SolvedEx.8
    For each of the following p.d.f. of r.v. XX, find (a) P(X<1)P(X < 1) and (b) P(∣X∣<1)P(|X| < 1) (i) f(x)=x218f(x) = \dfrac{x^2}{18}, for −3<x<3-3 < x < 3 and =0= 0, otherwise. (ii) f(x)=x+218f(x) = \dfrac{x + 2}{18}, for −2<x<4-2 < x < 4 and =0= 0, otherwise.
  9. 7.4 SolvedEx.9
    Find the c.d.f. F(x)F(x) associated with p.d.f. f(x)f(x) of r.v. XX where f(x)=3(1−2x2)f(x) = 3(1 - 2x^2), for 0<x<10 < x < 1 and =0= 0, otherwise.

Exercise 7.2

23 q
  1. Ex 7.2 Q.1 (i)
    Verify which of the following is p.d.f. of r.v. XX : f(x)=sin⁡xf(x) = \sin x, for 0≤x≤π20 \leq x \leq \dfrac{\pi}{2}
  2. Ex 7.2 Q.1 (ii)
    Verify which of the following is p.d.f. of r.v. XX : f(x)=xf(x) = x, for 0≤x≤10 \leq x \leq 1 and =2−x= 2 - x for 1<x<21 < x < 2
  3. Ex 7.2 Q.1 (iii)
    Verify which of the following is p.d.f. of r.v. XX : f(x)=2f(x) = 2, for 0≤x≤10 \leq x \leq 1
  4. Ex 7.2 Q.2 (a)
    The following is the p.d.f. of r.v. XX : f(x)=x8f(x) = \dfrac{x}{8}, for 0<x<40 < x < 4 and =0= 0 otherwise Find P(x<1.5)P(x < 1.5)
  5. Ex 7.2 Q.2 (b)
    The following is the p.d.f. of r.v. XX : f(x)=x8f(x) = \dfrac{x}{8}, for 0<x<40 < x < 4 and =0= 0 otherwise Find P(1<x<2)P(1 < x < 2)
  6. Ex 7.2 Q.2 (c)
    The following is the p.d.f. of r.v. XX : f(x)=x8f(x) = \dfrac{x}{8}, for 0<x<40 < x < 4 and =0= 0 otherwise Find P(x>2)P(x > 2)
  7. Ex 7.2 Q.3 (i)
    It is known that error in measurement of reaction temperature (in 0∘0^\circ c) in a certain experiment is continuous r.v. given by f(x)=x23f(x) = \dfrac{x^2}{3}, for −1<x<2-1 < x < 2 and =0= 0 otherwise Verify whether f(x)f(x) is p.d.f. of r.v. XX.
  8. Ex 7.2 Q.3 (ii)
    It is known that error in measurement of reaction temperature (in 0∘0^\circ c) in a certain experiment is continuous r.v. given by f(x)=x23f(x) = \dfrac{x^2}{3}, for −1<x<2-1 < x < 2 and =0= 0 otherwise Find P(0<x≤1)P(0 < x \leq 1)
  9. Ex 7.2 Q.3 (iii)
    It is known that error in measurement of reaction temperature (in 0∘0^\circ c) in a certain experiment is continuous r.v. given by f(x)=x23f(x) = \dfrac{x^2}{3}, for −1<x<2-1 < x < 2 and =0= 0 otherwise Find probability that XX is negative.
  10. Ex 7.2 Q.4 (i)
    Find kk if the following function represent p.d.f. of r.v. XX. f(x)=kxf(x) = kx, for 0<x<20 < x < 2 and =0= 0 otherwise, Also find P(14<x<32)P\left(\dfrac{1}{4} < x < \dfrac{3}{2}\right).
  11. Ex 7.2 Q.4 (ii)
    Find kk if the following function represent p.d.f. of r.v. XX. f(x)=kx(1−x)f(x) = kx(1 - x), for 0<x<10 < x < 1 and =0= 0 otherwise, Also find P(14<x<12)P\left(\dfrac{1}{4} < x < \dfrac{1}{2}\right), P(x<12)P\left(x < \dfrac{1}{2}\right).
  12. Ex 7.2 Q.5 (i)
    Let XX be amount of time for which a book is taken out of library by randomly selected student and suppose XX has p.d.f. f(x)=0.5xf(x) = 0.5x, for 0≤x≤20 \leq x \leq 2 and =0= 0 otherwise. Calculate P(X≤1)P(X \leq 1)
  13. Ex 7.2 Q.5 (ii)
    Let XX be amount of time for which a book is taken out of library by randomly selected student and suppose XX has p.d.f. f(x)=0.5xf(x) = 0.5x, for 0≤x≤20 \leq x \leq 2 and =0= 0 otherwise. Calculate P(0.5≤x≤1.5)P(0.5 \leq x \leq 1.5)
  14. Ex 7.2 Q.5 (iii)
    Let XX be amount of time for which a book is taken out of library by randomly selected student and suppose XX has p.d.f. f(x)=0.5xf(x) = 0.5x, for 0≤x≤20 \leq x \leq 2 and =0= 0 otherwise. Calculate P(x≥1.5)P(x \geq 1.5)
  15. Ex 7.2 Q.6 (i)
    Suppose that XX is waiting time in minutes for a bus and its p.d.f. is given by f(x)=15f(x) = \dfrac{1}{5}, for 0≤x≤50 \leq x \leq 5 and =0= 0 otherwise. Find the probability that waiting time is between 1 and 3
  16. Ex 7.2 Q.6 (ii)
    Suppose that XX is waiting time in minutes for a bus and its p.d.f. is given by f(x)=15f(x) = \dfrac{1}{5}, for 0≤x≤50 \leq x \leq 5 and =0= 0 otherwise. Find the probability that waiting time is more than 4 minutes.
  17. Ex 7.2 Q.7 (i)
    Suppose error involved in making a certain measurement is continuous r.v. XX with p.d.f. f(x)=k(4−x2)f(x) = k(4 - x^2), for −2≤x≤2-2 \leq x \leq 2 and =0= 0 otherwise. Compute : P(x>0)P(x > 0)
  18. Ex 7.2 Q.7 (ii)
    Suppose error involved in making a certain measurement is continuous r.v. XX with p.d.f. f(x)=k(4−x2)f(x) = k(4 - x^2), for −2≤x≤2-2 \leq x \leq 2 and =0= 0 otherwise. Compute : P(−1<x<1)P(-1 < x < 1)
  19. Ex 7.2 Q.7 (iii)
    Suppose error involved in making a certain measurement is continuous r.v. XX with p.d.f. f(x)=k(4−x2)f(x) = k(4 - x^2), for −2≤x≤2-2 \leq x \leq 2 and =0= 0 otherwise. Compute : P(−0.5<x or x>0.5)P(-0.5 < x \text{ or } x > 0.5)
  20. Ex 7.2 Q.8 (i)
    The following is the p.d.f. of continuous r.v. f(x)=x8f(x) = \dfrac{x}{8}, for 0<x<40 < x < 4 and =0= 0 otherwise. Find expression for c.d.f. of XX
  21. Ex 7.2 Q.8 (ii)
    The following is the p.d.f. of continuous r.v. f(x)=x8f(x) = \dfrac{x}{8}, for 0<x<40 < x < 4 and =0= 0 otherwise. Find F(x)F(x) at x=0.5x = 0.5, 1.7 and 5.
  22. Ex 7.2 Q.9
    Given the p.d.f. of a continuous r.v. XX, f(x)=x23f(x) = \dfrac{x^2}{3}, for −1<x<2-1 < x < 2 and =0= 0 otherwise Determine c.d.f. of XX hence find P(x<1)P(x < 1), P(x<−2)P(x < -2), P(X>0)P(X > 0), P(1<x<2)P(1 < x < 2)
  23. Ex 7.2 Q.10
    If a r.v. XX has p.d.f., f(x)=cxf(x) = \dfrac{c}{x}, for 1<x<31 < x < 3, c>0c > 0, Find cc, E(X)\mathrm{E}(X) and Var(X)\mathrm{Var}(X).

Miscellaneous Exercise 7

25 q

Choose the correct option

Practice · 10
  1. Misc 7 I (1)
    P.d.f. of a.c.r.v XX is f(x)=6x(1−x)f(x) = 6x(1 - x), for 0≤x≤10 \leq x \leq 1 and =0= 0, otherwise (elsewhere) If P(X<a)=P(X>a)P(X < a) = P(X > a), then a=a =
    1. A.
      11
    2. B.
      12\dfrac{1}{2}
    3. C.
      13\dfrac{1}{3}
    4. D.
      14\dfrac{1}{4}
  2. Misc 7 I (2)
    If the p.d.f of a.c.r.v. XX is f(x)=3(1−2x2)f(x) = 3(1 - 2x^{2}), for 0<x<10 < x < 1 and =0= 0, otherwise (elsewhere) then the c.d.f of XX is F(x)=F(x) =
    1. A.
      2x−3x22x - 3x^{2}
    2. B.
      3x−4x33x - 4x^{3}
    3. C.
      3x−2x33x - 2x^{3}
    4. D.
      2x3−3x2x^{3} - 3x
  3. Misc 7 I (3)
    If the p.d.f of a.c.r.v. XX is f(x)=x218f(x) = \dfrac{x^{2}}{18}, for −3<x<3-3 < x < 3 and =0= 0, otherwise then P(∣X∣<1)=P\left(\left|X\right| < 1\right) =
    1. A.
      127\dfrac{1}{27}
    2. B.
      128\dfrac{1}{28}
    3. C.
      129\dfrac{1}{29}
    4. D.
      126\dfrac{1}{26}
  4. Misc 7 I (4)
    If a d.r.v. XX takes values 0,1,2,3,…0, 1, 2, 3, \ldots which probability P(X=x)=k(x+1)⋅5−xP(X = x) = k(x + 1) \cdot 5^{-x}, where k is a constant, then P(X=0)=P(X = 0) =
    1. A.
      725\dfrac{7}{25}
    2. B.
      1625\dfrac{16}{25}
    3. C.
      1825\dfrac{18}{25}
    4. D.
      1925\dfrac{19}{25}
  5. Misc 7 I (5)
    If p.m.f. of a d.r.v. XX is P(X=x)=(5Cx)25P(X = x) = \dfrac{\left({}^{5}C_{x}\right)}{2^{5}}, for x=0,1,2,3,4,5x = 0, 1, 2, 3, 4, 5 and =0= 0, otherwise If a=P(X≤2)a = P(X \leq 2) and b=P(X≥3)b = P(X \geq 3), then E(X)=E(X) =
    1. A.
      a<ba < b
    2. B.
      a>ba > b
    3. C.
      a=ba = b
    4. D.
      a+ba + b
  6. Misc 7 I (6)
    If p.m.f. of a d.r.v. XX is P(X=x)=xn(n+1)P(X = x) = \dfrac{x}{n(n + 1)}, for x=1,2,3,…,nx = 1, 2, 3, \ldots, n and =0= 0, otherwise then E(X)=E(X) =
    1. A.
      n1+12\dfrac{n}{1} + \dfrac{1}{2}
    2. B.
      n3+16\dfrac{n}{3} + \dfrac{1}{6}
    3. C.
      n2+15\dfrac{n}{2} + \dfrac{1}{5}
    4. D.
      n1+13\dfrac{n}{1} + \dfrac{1}{3}
  7. Misc 7 I (7)
    If p.m.f. of a d.r.v. XX is P(x)=cx3P(x) = \dfrac{c}{x^{3}}, for x=1,2,3x = 1, 2, 3 and =0= 0, otherwise (elsewhere) then E(X)=E(X) =
    1. A.
      343297\dfrac{343}{297}
    2. B.
      294251\dfrac{294}{251}
    3. C.
      297294\dfrac{297}{294}
    4. D.
      294297\dfrac{294}{297}
  8. Misc 7 I (8)
    If the a d.r.v. XX has the following probability distribution :
    XX−2-2−1-100112233
    P(X=x)P(X = x)0.1kk0.22k2k0.3kk
    then P(X=−1)=P(X = -1) =
    1. A.
      110\dfrac{1}{10}
    2. B.
      210\dfrac{2}{10}
    3. C.
      310\dfrac{3}{10}
    4. D.
      410\dfrac{4}{10}
  9. Misc 7 I (9)
    If the a d.r.v. XX has the following probability distribution :
    XX11223344556677
    P(X=x)P(X = x)kk2k2k2k2k3k3kk2k^{2}2k22k^{2}7k2+k7k^{2} + k
    then k=k =
    1. A.
      17\dfrac{1}{7}
    2. B.
      18\dfrac{1}{8}
    3. C.
      19\dfrac{1}{9}
    4. D.
      110\dfrac{1}{10}
  10. Misc 7 I (10)
    The expected value of XX for the following p.m.f.
    XX−2-2−1-1001122
    P(X)P(X)0.30.30.10.050.25
    1. A.
      0.850.85
    2. B.
      −0.35-0.35
    3. C.
      0.150.15
    4. D.
      −0.15-0.15

Solve the following

Practice · 15
  1. Misc 7 II (1)
    Identify the random variable as either discrete or continuous in each of the following. Write down the range of it. (i) An economist is interested the number of unemployed graduate in the town of population 1 lakh. (ii) Amount of syrup prescribed by physician. (iii) The person on the high protein diet is interested gain of weight in a week. (iv) 20 white rats are available for an experiment. Twelve rats are male. Scientist randomly selects 5 rats number of female rats selected on a specific day. (v) A highway safety group is interested in studying the speed (km/hrs) of a car at a check point.
  2. Misc 7 II (2)
    The probability distribution of discrete r.v. XX is as follows
    X=xX = x112233445566
    P(X=x)P(X = x)kk2k2k3k3k4k4k5k5k6k6k
    (i) Determine the value of kk. (ii) Find P(X≤4)P(X \leq 4), P(2<X<4)P(2 < X < 4), P(X≥3)P(X \geq 3).
  3. Misc 7 II (3)
    The following probability distribution of r.v. XX
    X=xX = x−3-3−2-2−1-100112233
    P(X=x)P(X = x)0.050.10.150.200.250.150.1
    Find the probability that (i) XX is positive. (ii) XX is non negative. (iii) XX is odd. (iv) XX is even.
  4. Misc 7 II (4)
    The p.m.f. of a r.v. X is given by P(X=x)=(5Cx)25P(X = x) = \dfrac{\left({}^{5}C_{x}\right)}{2^{5}}, for x=0,1,2,3,4,5x = 0, 1, 2, 3, 4, 5 and =0= 0, otherwise. Then show that P(X≤2)=P(X≥3)P(X \leq 2) = P(X \geq 3).
  5. Misc 7 II (5)
    In the p.m.f. of r.v. XX
    xx1122334455
    P(X)P(X)120\dfrac{1}{20}320\dfrac{3}{20}aa2a2a120\dfrac{1}{20}
    Find aa and obtain c.d.f. of XX.
  6. Misc 7 II (6)
    A fair coin is tossed 4 times. Let XX denotes the number of heads obtained write down the probability distribution of XX. Also find the formula for p.m.f. of XX.
  7. Misc 7 II (7)
    Find the probability distribution of the number of successes in two tosses of a die, where a success is defined as (i) number greater than 4 (ii) six appears on at least one die.
  8. Misc 7 II (8)
    A random variable XX has the following probability distribution.
    XX0011223344556677
    P(X)P(X)00kk2k2k2k2k3k3kk2k^{2}2k22k^{2}7k2+k7k^{2} + k
    Determine (i) kk (ii) P(X>6)P(X > 6) (iii) P(0<X<3)P(0 < X < 3)
  9. Misc 7 II (9)
    The following is the c.d.f. of r.v. XX
    XX−3-3−2-2−1-10011223344
    F(X)F(X)0.10.30.50.650.750.850.91
    Find p.m.f. of XX. (i) P(−1≤X≤2)P(-1 \leq X \leq 2) (ii) P(X≤3/X>0)P(X \leq 3 / X > 0)
  10. Misc 7 II (10)
    Find the expected value, variance and standard deviation of r.v. XX whose p.m.f. are given below. (i)
    X=xX = x112233
    P(X=x)P(X = x)15\dfrac{1}{5}25\dfrac{2}{5}25\dfrac{2}{5}
    (ii)
    X=xX = x−1-10011
    P(X=x)P(X = x)15\dfrac{1}{5}25\dfrac{2}{5}25\dfrac{2}{5}
    (iii)
    X=xX = x112233…\ldotsnn
    P(X=x)P(X = x)1n\dfrac{1}{n}1n\dfrac{1}{n}1n\dfrac{1}{n}…\ldots1n\dfrac{1}{n}
    (iv)
    X=xX = x001122334455
    P(X=x)P(X = x)132\dfrac{1}{32}532\dfrac{5}{32}1032\dfrac{10}{32}1032\dfrac{10}{32}532\dfrac{5}{32}132\dfrac{1}{32}
  11. Misc 7 II (11)
    A player tosses two coins he wins ₹ 10 if 2 heads appears, ₹ 5 if 1 head appears and ₹ 2 if no head appears. Find the expected winning amount and variance of winning amount.
  12. Misc 7 II (12)
    Let the p.m.f. of r.v. XX be P(x)=3−x10P(x) = \dfrac{3 - x}{10}, for x=−1,0,1,2x = -1, 0, 1, 2 and =0= 0, otherwise Calculate E(X)E(X) and Var(X)Var(X).
  13. Misc 7 II (13)
    Suppose error involved in making a certain measurement is continuous r.v. XX with p.d.f. f(x)=k(4−x2)f(x) = k(4 - x^{2}), for −2≤x≤2-2 \leq x \leq 2 and =0= 0 otherwise. Compute (i) P(X>0)P(X > 0) (ii) P(−1<x<1)P(-1 < x < 1) (iii) P(X<0.5P(X < 0.5 or X>0.5)X > 0.5)
  14. Misc 7 II (14)
    The p.d.f. of r.v. XX is given by f(x)=12af(x) = \dfrac{1}{2a}, for 0<x<2a0 < x < 2a and =0= 0, otherwise. Show that P(X<a2)=P(X>3a2)P\left(X < \dfrac{a}{2}\right) = P\left(X > \dfrac{3a}{2}\right).
  15. Misc 7 II (15)
    The p.d.f. of r.v. of XX is given by f(x)=kxf(x) = \dfrac{k}{\sqrt{x}}, for 0<x<40 < x < 4 and =0= 0, otherwise. Determine kk. Determine c.d.f. of XX and hence P(X≤2)P(X \leq 2) and P(X≤1)P(X \leq 1).