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Mean and variance of a random variable Maths - Mean and variance of a random variable

Maharashtra Board - STD 12 - Maths (MSBSHSE - Semi English Medium). 82 questions in this chapter.

Question types

Mean and variance of a random variable question types

82 questions across 4 question groups — pick any mix to generate a Maths paper with step-by-step answer keys.

82
Questions
4
Question groups
5
Question types
Sample Questions

Mean and variance of a random variable questions

One sample from each question group in this chapter. Select any group above to see the full set with answer keys.

Q 1MCQ1 Mark
The probability distribution of a discrete random variable $X$ is given below:
$\text{X}:$
$1$
$2$
$3$
$4$
$\text{P}(\text{X}):$
$\frac{1}{10}$
$\frac{1}{5}$
$\frac{3}{10}$
$\frac{2}{5}$
The value of $E(X^2)$ is:
  • A
    $3$
  • B
    $5$
  • C
    $7$
  • ✓
    $10$

Answer: D.

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Q 2MCQ1 Mark
The probability distribution of a discrete random variable $X$ is given below:
$\text{X}:$ $2$ $3$ $4$ $5$
$\text{P}(\text{X}):$ $\frac{5}{\text{k}}$ $\frac{7}{\text{k}}$ $\frac{9}{\text{k}}$ $\frac{11}{\text{k}}$
The value of $k$ is:
  • A
    $8$
  • B
    $16$
  • ✓
    $32$
  • D
    $48$

Answer: C.

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Q 3MCQ1 Mark
Let $X$ be a discrete random variable. Then the variance of $X$ is:
  • A
    $E(X^2)$
     
  • B
    $E(X^2) + (E(X))^2$
     
  • ✓
    $E(X^2) - (E(X))^2$
     
  • D
    $\sqrt{\text{E}(\text{X}^2)-(\text{E}(\text{X}))^2}$

Answer: C.

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Q 4MCQ1 Mark
For the following probability distribution:
$X:$ $-4$ $-3$ $-2$ $-1$ $0$
$P(X):$ $0.1$ $0.2$ $0.3$ $0.2$ $0.2$
The value of $E(X)$ is:
  • A
    $0$
  • B
    $-1$
  • C
    $-2$
  • ✓
    $-1.8$

Answer: D.

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Q 5MCQ1 Mark
If $X$ is a random variable with probability distribution as given below:
$X = x_i$ $0$ $1$ $2$ $3$
$P(X = X_i)$ $k$ $3k$ $3k$ $k$
The value of $k$ and its variance are:
  • A
    $\frac{1}{8},\frac{22}{27}$
  • B
    $\frac{1}{8},\frac{23}{27}$
  • C
    $\frac{1}{8},\frac{24}{27}$
  • ✓
    $\frac{1}{8},\frac{3}{4}$

Answer: D.

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Q 13Solve the Following Question.(3 Marks)3 Marks
Find the mean and standard deviation of the following probability distributions:
$\text{x}_\text{i}$
$0$
$1$
$2$
$3$
$4$
$5$
$\text{p}_\text{i}$
$\frac{1}{6}$
$\frac{5}{18}$
$\frac{2}{9}$
$\frac{1}{6}$
$\frac{1}{9}$
$\frac{1}{18}$
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Q 16Solve the Following Question.(5 Marks)5 Marks
Two cards are selected at random from a box which contains five cards numbered $1, 1, 2, 2,$ and $3$. Let X denote the sum and Y the maximum of the two numbers drawn. Find the probability distribution, mean and variance of X and Y.
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Q 18Solve the Following Question.(5 Marks)5 Marks
In a game, a man wins Rs 5 for getting a number greater than 4 and loses Rs 1 otherwise, when a fair die is thrown. The man decided to thrown a die thrice but to quit as and when he gets a number greater then 4. Find the expected value of the amount he wins or loses.
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Q 19Solve the Following Question.(5 Marks)5 Marks
Let X denot the number of colleges where you will apply after your results and P(X = x) denotes your probability of getting admission in x number of colleges. It is given that
$\text{P}(\text{X = x})=\begin{cases}\text{kx},&\text{if}\text{ x}=0\text{ or }1\\2\text{kx},&\text{if x = 2}\\\text{k}(5-\text{x}),&\text{if x = 3 or 4}\\0,&\text{if x > 4}\end{cases}$
where k is a positive constant. Find the value of k. Also find the probability that you will get addmission in
  1. Exactly one college.
  2. At most two colleges.
  3. At least two colleges.
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