Probability distribution of a random variable — practice questions

Question 1 · difficulty L2 · understanding

A bag contains 4 white and 6 black balls. Three balls are drawn at random from the bag. Let $$\mathrm{X}$$ be the number of white balls, among the drawn balls. If $$\sigma^{2}$$ is the variance of $$\mathrm{X}$$, then $$100 \sigma^{2}$$ is equal to ________.

Question 2 · difficulty L2 · understanding

The probability distribution of random variable X is given by : X 1 2 3 4 5 P(X) K 2K 2K 3K K Let p = P(1 < X < 4 | X < 3). If 5p = $$\lambda$$K, then $$\lambda$$ equal to ___________.

Question 3 · difficulty L2 · understanding

The probability distribution of a random variable X is given below : X 4k $\frac{30}{7}k$ $\frac{32}{7}k$ $\frac{34}{7}k$ $\frac{36}{7}k$ $\frac{38}{7}k$ $\frac{40}{7}k$ 6k P(X) $\frac{2}{15}$ $\frac{1}{15}$ $\frac{2}{15}$ $\frac{1}{5}$ $\frac{1}{15}$ $\frac{2}{15}$ $\frac{1}{5}$ $\frac{1}{15}$ If E(X) = $\frac{263}{15}$, then P(X < 20) is equal to :

• A. $\frac{3}{5}$
• B. $\frac{14}{15}$
• C. $\frac{8}{15}$
• D. $\frac{11}{15}$

Question 4 · difficulty L2 · understanding

If a random variable $x$ has the probability distribution $$\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \mathrm{P}(x) & 0 & 2 \mathrm{k} & \mathrm{k} & 3 \mathrm{k} & 2 \mathrm{k}^2 & 2 \mathrm{k} & \mathrm{k}^2+\mathrm{k} & 7 \mathrm{k}^2 \\ \hline \end{array}$$ $$\text { then } \mathrm{P}(3 < x \leq 6) \text { is equal to }$$

• A. 0.34
• B. 0.64
• C. 0.22
• D. 0.33

Question 5 · difficulty L2 · understanding

A random variable X has the following probability distribution : X 0 1 2 3 4 P(X) k 2k 4k 6k 8k The value of P(1 < X < 4 | X $$\le$$ 2) is equal to :

• A. $${4 \over 7}$$
• B. $${2 \over 3}$$
• C. $${3 \over 7}$$
• D. $${4 \over 5}$$

Question 6 · difficulty L3 · understanding

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables $$X$$ and $$Y$$ respectively denote the number of blue and yellow balls. If $$\bar{X}$$ and $$\bar{Y}$$ are the means of $$X$$ and $$Y$$ respectively, then $$7 \bar{X}+4 \bar{Y}$$ is equal to ___________.

Question 7 · difficulty L3 · understanding

From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable $$X$$ denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of $$X$$ is $$\frac{m}{n}$$, where $$\operatorname{gcd}(m, n)=1$$, then $$n-m$$ is equal to _________.

Question 8 · difficulty L3 · understanding

From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable $$X$$ denote the number of defective items in the sample. If the variance of $$X$$ is $$\sigma^2$$, then $$96 \sigma^2$$ is equal to __________.

Question 9 · difficulty L3 · understanding

Let the probability of getting head for a biased coin be $$\frac{1}{4}$$. It is tossed repeatedly until a head appears. Let $$\mathrm{N}$$ be the number of tosses required. If the probability that the equation $$64 \mathrm{x}^{2}+5 \mathrm{Nx}+1=0$$ has no real root is $$\frac{\mathrm{p}}{\mathrm{q}}$$, where $$\mathrm{p}$$ and $$\mathrm{q}$$ are coprime, then $$q-p$$ is equal to ________.

Question 10 · difficulty L3 · understanding

The probability distribution of X is : X 0 1 2 3 P(X) $${{1 - d} \over 4}$$ $${{1 + 2d} \over 4}$$ $${{1 - 4d} \over 4}$$ $${{1 + 3d} \over 4}$$ For the minimum possible value of d, sixty times the mean of X is equal to _______________.