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Mean and variance of a random variable question types

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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.

If the random variable X has the following distribution:
X: 0 1 2 3 4 5 6 7 8
P(X): a 3a 5a 7a 9a 11a 13a 15a 17a
then the value of a is:
  1. $\frac{7}{81}$
  2. $\frac{5}{81}$
  3. $\frac{2}{81}$
  4. $\frac{1}{81}$
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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:
  1. 8
  2. 16
  3. 32
  4. 48
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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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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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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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The probability distribution of random variable X is given below:
$\text{X}$
$0$
$1$
$2$
$3$
$\text{P}(\text{X})$
$\text{k}$
$\frac{\text{k}}{2}$
$\frac{\text{k}}{4}$
$\frac{\text{k}}{8}$
 Find $\text{P}(\text{X}\leq2)+\text{P}(\text{X}>2)$
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Find the mean of the following probability distribution:
$\text{X}=\text{x}_\text{i}:$ $1$ $2$ $3$
$\text{P}(\text{X}=\text{x}_\text{i}):$ $\frac{1}{4}$ $\frac{1}{8}$ $\frac{5}{8}$
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The probability distribution function oif a random variable $X$ is given by
$X_i$ $0$ $1$ $2$
$P_i$ $3c^3$ $4c - 10c^2$ $5c - 1$
Where $c > 0$
Find: $P(X < 2).$
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The probability distribution function oif a random variable $X$ is given by
$X_i$ $0$ $1$ $2$
$P_i$ $3c^3$ $4c - 10c^2$ $5c - 1$
Where $c > 0$
Find: $c.$
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The probability distribution of random variable X is given below:
$\text{X}$
$0$
$1$
$2$
$3$
$\text{P}(\text{X})$
$\text{k}$
$\frac{\text{k}}{2}$
$\frac{\text{k}}{4}$
$\frac{\text{k}}{8}$
 Determine $\text{P}(\text{X}\leq2)$ and $\text{P}(\text{X}>2)$
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The probability distribution of random variable X is given below:
$\text{X}$
$0$
$1$
$2$
$3$
$\text{P}(\text{X})$
$\text{k}$
$\frac{\text{k}}{2}$
$\frac{\text{k}}{4}$
$\frac{\text{k}}{8}$
 Determine the value of k.
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Q 123 Marks Question3 Marks
An urn contain 5 red and 2 black balls. Two balls rendomly selected. Let X represent the number of black ball. What are the possible values of X. Is X a random variable?
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Q 133 Marks Question3 Marks
A fair die is tossed twice. If the number appearing on the top is less than 3, it is success. Find the probability distribution of number of successes.
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Q 153 Marks Question3 Marks
Five defective bolts are acciedently mixed with twenty good ones. If four bolts are drawn at random from this lot, find the probability distribution of the number of defective bolts.
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Q 163 Marks Question3 Marks
Let $X$ be a random variable which assumes values $x_1, x_2, x_3, x_4$ such that $2P(X = x_1) = 3P(X = x_2) = P(X = x_3) = 5P(X = x_4).$ Find the probability distribution of $X.$
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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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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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Two numbers are selected at random (without replacement) from positive integers 2, 3, 4, 5, 6 and 7. Let X denote the larger of the two numbers obtained. Find the mean and variance of the probability distribution of X.
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A class has 15 students whose ages are 14, 17, 15, 14, 21, 19, 20, 16, 18, 17, 20, 17, 16, 19 and 20 years respectively. One student is selected in such a manner that each has the same chance to being selected and the age X of the selected student is recorded. What is the probability distribution of the random variable X.
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