Find the expected value of X for the following p.m.f.
- A0.85
- ✓-0.35
- C0.15
- D-0.15
Answer: B.
View full solution →Question types
Probability Distributions question types
185 questions across 6 question groups — pick any mix to generate a Maths paper with step-by-step answer keys.
185
Questions
6
Question groups
5
Question types
01
MCQ
101 Q→02Answer the following questions in short.
29 Q→03Solve the Following Question.(2 Marks)
12 Q→04Solve the Following Question.(3 Marks)
24 Q→05Solve the Following Question.(4 Marks)
13 Q→06Solve the Following Question.(5 Marks)
6 Q→Sample Questions
Probability Distributions questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If the d.r.v. X has the following probability distribution:
then k =
then k =
- A$\frac{1}{7}$
- B$\frac{1}{8}$
- C$\frac{1}{9}$
- ✓$\frac{1}{10}$
Answer: D.
View full solution →If the d.r.v. X has the following probability distribution:
then P(X = -1) =
then P(X = -1) =
- ✓$\frac{1}{10}$
- B$\frac{2}{10}$
- C$\frac{3}{10}$
- D$\frac{4}{10}$
Answer: A.
View full solution →If p.m.f. of a d.r.v. $\mathrm{X}$ is $P(x)=\frac{c}{x^3}$, for $x=1,2,3$ and $=0$, otherwise (elsewhere), then $E(X)=$
- A$\frac{343}{297}$
- ✓$\frac{294}{251}$
- C$\frac{297}{294}$
- D$\frac{294}{297}$
Answer: B.
View full solution →If p.m.f. of a d.r.v. $\mathrm{X}$ is $\mathrm{P}(\mathrm{X}=\mathrm{x})=\frac{x}{n(n+1)}$, for $\mathrm{x}=1,2,3, \ldots \ldots, \mathrm{n}$ and $=0$, otherwise, then $\mathrm{E}(\mathrm{X})$ $=$
- A$\frac{n}{1}+\frac{1}{2}$
- ✓$\frac{n}{3}+\frac{1}{6}$
- C$\frac{n}{2}+\frac{1}{5}$
- D$\frac{n}{1}+\frac{1}{3}$
Answer: B.
View full solution →Find the c.d.f. $F(x)$ associated with p.d.f. $f(x)$ of r.v. $X$ where $f(x)=3\left(1-2 x^2\right) \quad$ for $0<x<1$ and $=0$, otherwise.
View full solution →Find $k$ if the following function is the p.d.f. of r.v. $X$.
$
f(x)=k x^2(1-x), \quad \text { for } 0<x<1 \text { and }=0 \text {, otherwise. }
$
View full solution →$
f(x)=k x^2(1-x), \quad \text { for } 0<x<1 \text { and }=0 \text {, otherwise. }
$
Verify if the following functions are p.d.f. of a continuous r.v. $X$. : $f(x)=\frac{x}{2}$, for $-2<x<2$ and $=0$, otherwise.
View full solution →Verify if the following functions are p.d.f. of a continuous r.v. $X$. : $f(x)=e^{-x}$, for $0<x<\infty$ and $=0$, otherwise.
View full solution →Let's return to the example in which $X$ has the following probability density function :
$f(x)=\frac{x^3}{4}$ for $0<x<4$. What is the cumulative distribution function $X$ ?
View full solution →$f(x)=\frac{x^3}{4}$ for $0<x<4$. What is the cumulative distribution function $X$ ?
The p.m.f. of a r.v. $X$ is given by $P(X=x)=\frac{{ }^5 C_x}{2^5}$, for $x=0,1,2,3,4,5$ and $=0$, otherwise. Then show that $P(X \leq 2)=P(X \geq 3)$.
View full solution →For each of the following p.d.f. of r.v. $X$, find (a) $P(X<1)$ and (b) $P(|X|<1)$ : $f(x)=\frac{x+2}{18}$, for $-2<x<4$ and $=0$, otherwise.
View full solution →For each of the following p.d.f. of r.v. $X$, find (a) $P(X<1)$ and (b) $P(|X|<1)$ : $f(x)=\frac{x^2}{18}, \quad$ for $-3<x<3$ and $=0$, otherwise.
View full solution →Let X be the amount of time for which a book is taken out of the library by a randomly selected students and suppose X has p.d.f.
f(x) = 0.5x, for 0 ≤ x ≤ 2 and = 0 otherwise.
Calculate:
P(x ≥ 1.5).
f(x) = 0.5x, for 0 ≤ x ≤ 2 and = 0 otherwise.
Calculate:
P(x ≥ 1.5).
Let X be the amount of time for which a book is taken out of the library by a randomly selected students and suppose X has p.d.f.
f(x) = 0.5x, for 0 ≤ x ≤ 2 and = 0 otherwise.
Calculate:
P(0.5 ≤ x ≤ 1.5)
f(x) = 0.5x, for 0 ≤ x ≤ 2 and = 0 otherwise.
Calculate:
P(0.5 ≤ x ≤ 1.5)
The p.d.f. of r.v. $X$ is given by $f ( x )=\frac{k}{\sqrt{x}}$, for $0< x <4$ and $=0$, otherwise. Determine $k$. Determine c.d.f. of $X$ and hence find $P(X \leq 2)$ and $P(X \leq 1)$.
View full solution →The p.d.f. of a continuous r.v. $X$ is given by $f ( x )=\frac{1}{2 a}$, for $0< x <2 a$ and $=0$, otherwise. Show that $P \left( X <\frac{a}{2}\right)= P \left( X >\frac{3 a}{2}\right)$
View full solution →Find the expected value, variance, and standard deviation of the random variable whose p.m.f’s are given below:
Find the expected value, variance, and standard deviation of the random variable whose p.m.f’s are given below:
Find the expected value, variance, and standard deviation of the random variable whose p.m.f’s are given below:
Suppose the error involved in making a certain measurement is a continuous r.v. X with p.d.f.
$f(x) = k(4 – x^2), -2 \leq x \leq 2$ and $= 0$ otherwise.
Compute
$P(X < -0.5$ or $X > 0.5)$
View full solution →$f(x) = k(4 – x^2), -2 \leq x \leq 2$ and $= 0$ otherwise.
Compute
$P(X < -0.5$ or $X > 0.5)$
Suppose the error involved in making a certain measurement is a continuous r.v. X with p.d.f.
$f(x) = k(4 – x^2), -2 \leq x \leq 2$ and $= 0$ otherwise.
Compute
$P(-1 < X < 1)$
View full solution →$f(x) = k(4 – x^2), -2 \leq x \leq 2$ and $= 0$ otherwise.
Compute
$P(-1 < X < 1)$
Suppose the error involved in making a certain measurement is a continuous r.v. X with p.d.f.
$f(x) = k(4 – x^2), -2 \leq x \leq 2$ and $= 0$ otherwise.
Compute
$P(X > 0)$
View full solution →$f(x) = k(4 – x^2), -2 \leq x \leq 2$ and $= 0$ otherwise.
Compute
$P(X > 0)$
Let the p.m.f. of r.v. $X$ be $P(x)=\frac{3-x}{10}$, for $x =-1,0,1,2$ and $=0$, otherwise. Calculate $E(X)$ and $\operatorname{Var}(X)$.
View full solution →A player tosses two wins. He wins ₹ 10 if 2 heads appear, ₹ 5 if 1 head appears and ₹ 2 if no head appears. Find the expected winning amount and variance of the winning amount.
Identify the random variable as either discrete or continuous in each of the following. If the random variable is discrete, list its possible values:
(i) An economist is interested in the number of unemployed graduates in the town of population 1 lakh.
(ii) Amount of syrup prescribed by a physician.
(iii) The person on a high protein diet is interesting to gain weight in a week.
(iv) 20 white rats are available for an experiment. Twelve rats are males. A scientist randomly selects 5 rats, the number of female rats selected on a specific day.
(v) A highway-safety group is interested in studying the speed (in km/hr) of a car at a checkpoint.
(i) An economist is interested in the number of unemployed graduates in the town of population 1 lakh.
(ii) Amount of syrup prescribed by a physician.
(iii) The person on a high protein diet is interesting to gain weight in a week.
(iv) 20 white rats are available for an experiment. Twelve rats are males. A scientist randomly selects 5 rats, the number of female rats selected on a specific day.
(v) A highway-safety group is interested in studying the speed (in km/hr) of a car at a checkpoint.
Suppose the p.d.f. of a continuous random variable $X$ is defined as:
$f(x)=x+1$, for $-1<x<0$, and $f(x)=1-x, \quad$ for $0 \leq x<1$.
Find the c.d.f. $F(x)$.
View full solution →$f(x)=x+1$, for $-1<x<0$, and $f(x)=1-x, \quad$ for $0 \leq x<1$.
Find the c.d.f. $F(x)$.
Let a pair of dice be thrown and the random variable $X$ be the sum of the numbers that appear on the two dice. Find the mean or expectation of $X$ and variance of $X$.
View full solution →Suppose the error involved in making a certain measurement is a continuous r.v. X with p.d.f.
f(x) = k(4 – x2), -2 ≤ x ≤ 2 and 0 otherwise.
Compute:
(i) P(X > 0)
(ii) P(-1 < X < 1)
(iii) P(-0.5 < X or X > 0.5).
f(x) = k(4 – x2), -2 ≤ x ≤ 2 and 0 otherwise.
Compute:
(i) P(X > 0)
(ii) P(-1 < X < 1)
(iii) P(-0.5 < X or X > 0.5).
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 X of the student is recorded. What is the probability distribution of the random variable X? Find mean, variance, and standard deviation of X.
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