The probability distribution of a random variable X is given belo… - Vidyadip

Question

The probability distribution of a 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.
Determine $\text{P}(\text{X}\leq2)$ and $\text{P}(\text{X}\geq2)$
Find $\text{P}(\text{X}\leq2)+\text{P}(\text{X}\geq2)$

✓

Answer

We have,

$\text{X}$	$0$	$1$	$2$	$3$
$\text{P}(\text{X})$	$\text{k}$	$\frac{\text{k}}{2}$	$\frac{\text{k}}{4}$	$\frac{\text{k}}{8}$

Since, $\sum_\limits{\text{i}=1}^\text{n}\text{P}_{\text{i}}=1,\text{i}=1,2, ....,\text{n}$ and $\text{P}_{\text{i}}\geq0$

$\therefore\text{k}+\frac{\text{k}}{2}+\frac{\text{k}}{4}+\frac{\text{k}}{8}=1$

$\Rightarrow8\text{k}+4\text{k}+2\text{k}+\text{k}=8$

$\therefore\text{k}=\frac{8}{15}$

$\text{P}(\text{X}\leq2)=\text{P}(0)+\text{P}(1)+\text{P}(2)$

$=\text{k}+\frac{\text{k}}{2}+\frac{\text{k}}{4}$

$=\frac{(4\text{k}+2\text{k}+\text{k})}{4}=\frac{7\text{k}}{4}$

$=\frac{7}{4}\cdot\frac{8}{15}=\frac{14}{15}$

And $\text{P}(\text{X}\geq2)=\text{P}(3)=\frac{\text{k}}{8}$

$=\frac{1}{8}\cdot\frac{8}{15}=\frac{1}{15}$

$\text{P}(\text{X}\leq2)+\text{P}(\text{X}\geq2)$

$=\frac{14}{15}+\frac{1}{15}=1$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from Probability→Probability — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

The vertices A, B, C of triangle ABC have respectively position vector $\vec{\text{a}},\ \vec{\text{b}},\ \vec{\text{c}}$ with respect to given origin O. Show that the point D where the bisector of $\angle{\text{A}}$ meets BC has position Vector $\vec{\text{d}}=\frac{\beta\vec{\text{b}}+\gamma\vec{\text{c}}}{\beta+\gamma}$, where $\beta=\big|\vec{\text{c}}-\vec{\text{a}}\big|$ and, $\gamma=\big|\vec{\text{a}}-\vec{\text{b}}\big|$.

Evaluate definite integral $\int_0^\pi \frac{x \tan x}{\sec x+\tan x} d x$.

Give an example of a function:
Which is not one-one but onto.

If log y = tan^–1 x, then show that $\text{(1 + x}^{2}) \frac{\text{d}^{2}\text{y}}{\text{dx}^{2}} + \text{(2x + 1)} \frac{\text{dy}}{\text{dx}} = 0.$

A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of

17.50 per package on nuts and

7 per package of bolts. How many packages of each should be produced each day so as to maximise his profits if he operates his machines for at the most 12 hours a day? Form the above as a linear programming problem and solve it graphically.

Find the equation of the plane passing through the points (-1, 2, 0), (2, 2, -1) and parallel to the line $\frac{\text{x}-1}{1}=\frac{2\text{y}+1}{2}=\frac{\text{z}+1}{-1}.$

An aeroplane can carry a maximum of 200 passengers. A profit of Rs.1000 is made on each executive class ticket and a profit of Rs.600 is made on each economy class ticket. The airline reserves atleast 20 seats for executive class. However, atleast 4 times as many passengers prefer to travel by economy class than by the executive class. Determine how many tickets of each type must be sold in order to maximise the profit of the airline. What is the maximum profit?

Evaluate the following integrals:
$\int\limits^{\frac{\pi}{2}}_0\frac{\sqrt{\cot\text{x}}}{\sqrt{\cot\text{x}}+\sqrt{\tan\text{x}}}\text{ dx}$

Minimise Z = x + 2y
subject to $2\text{x}+\text{y}\geq3,\ \text{x}+2\text{y}\geq6,\ \text{x},\ \text{y}\geq0.$
Show that the minimum of Z occurs at more than two points.

An experiment succeeds twice as often as it fails. Find the probability that in the next 6 trials there will be at least 4 successes.