Applied Mathematics

Probability Distribution

Medium

applied

If the random variable X follows the Poisson distribution such that P[X = k] = P[X = k+1], then the mean value of X is:

k - 1

k²

k + 1

✅ Correct Option: 4

For a Poisson distribution where $P[X = k] = P[X = k+1]$ , the mean value of $X$ needs to be determined.

The probability formula for a Poisson distribution is:

$P[X = n] = \dfrac{\lambda^n \times e^{-\lambda}}{n!}$

where $\lambda$ is the mean of the distribution.

Given that $P[X = k] = P[X = k+1]$ :

$P[X = k] = \dfrac{\lambda^k \times e^{-\lambda}}{k!}$

$P[X = k+1] = \dfrac{\lambda^{k+1} \times e^{-\lambda}}{(k+1)!}$

Setting them equal:

$\dfrac{\lambda^k \times e^{-\lambda}}{k!} = \dfrac{\lambda^{k+1} \times e^{-\lambda}}{(k+1)!}$

Canceling $e^{-\lambda}$ from both sides:

$\dfrac{\lambda^k}{k!} = \dfrac{\lambda^{k+1}}{(k+1)!}$

Since $(k+1)! = (k+1) \times k!$ :

$\dfrac{\lambda^k}{k!} = \dfrac{\lambda^{k+1}}{(k+1) \times k!}$

Canceling $k!$ :

$\lambda^k = \dfrac{\lambda^{k+1}}{k+1}$

$(k+1) \times \lambda^k = \lambda^{k+1}$

$(k+1) \times \lambda^k = \lambda \times \lambda^k$

Dividing both sides by $\lambda^k$ :

$k + 1 = \lambda$

Therefore, the mean value $\lambda = k + 1$

2026: 29 May Shift 1

The mean of the number of tails in three tosses of a coin is :

2026: 18 May Shift 2

A pair of dice is thrown 5 times. If getting a total of 7 is considered a success, then

Match the LIST-I with LIST-II

LIST-I	LIST-II
A. Probability of no success	I. $26\left(\dfrac{1}{6}\right)^5$
B. Probability of 5 successes	II. $2\left(\dfrac{5}{6}\right)^5$
C. Probability of at least 4 successes	III. $\left(\dfrac{1}{6}\right)^5$
D. Probability of at most one success	IV. $\left(\dfrac{5}{6}\right)^5$

Choose the correct answer from the options given below:

2026: 12 May Shift 2

The probability distribution of a random variable $X$ is given as $P(X=x)=\begin{cases}kx^3 & \text{for } x=1,2,3\\ 3kx & \text{for } x=4,5\\ 0 & \text{otherwise}\end{cases}$ , where $k$ is a constant.

The value of $P(X\ge4)$ is

2026: 14 May Shift 2

In a random experiment, a collection of trials is called Bernoulli trials if

A. The number of trials is infinite.

B. The trials are independent of each other.

C. Each trial has exactly two outcomes, defined as success and failure.

D. The probability of success will vary in each trial.

Choose the correct answer from the options given below:

2023: 30 May Shift 3

Which of the following statements are correct ?

(A) $\text{var}(aX + b) = a^2 \text{var}(X)$

(B) $\text{var}(X) = E(X^2) - \{E(X)\}^2$

(D) $E(X) = \sum_{i=1}^{n} p_i x_i^2$

Choose the correct answer from the options given below :

2023: 23 May Shift 3

A shopkeeper wants to check the average number of cars sold per call. Past record of sales is shown below:

Sale of cars (Units)	0	1	2	3
Probabilities	$\frac{1}{6}$	$\frac{1}{2}$	$\frac{3}{10}$	$\frac{1}{30}$

The expected number of cars sold is :

If the random variable X follows the Poisson distribution such that P[X = k] = P[X = k+1], then the mean value of X is:

Solution

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