(B) The probability density function is given by $f(x) = \frac{1}{5}$ for $0 \leq x \leq 5$ and $0$ otherwise.We need to find the probability that the

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(B) The probability density function is given by $f(x) = \frac{1}{5}$ for $0 \leq x \leq 5$ and $0$ otherwise.We need to find the probability that the waiting time $X$ is not more than $4$ minutes,which is $P(X \leq 4)$.This is calculated by integrating the probability density function from the lower bound to $4$:$P(X \leq 4) = \int_{0}^{4} f(x) \, dx$$P(X \leq 4) = \int_{0}^{4} \frac{1}{5} \, dx$$P(X \leq 4) = \left[ \frac{x}{5} \right]_{0}^{4}$$P(X \leq 4) = \frac{4}{5} - \frac{0}{5} = \frac{4}{5} = 0.8$Therefore,the probability is $0.8$.

Class 12 Mathematics Probability Probability distribution

Difficult

If random variable $X$ is the waiting time in minutes for a bus and the probability density function of $X$ is given by $f(x) = \begin{cases} \frac{1}{5}, & 0 \leq x \leq 5 \\ 0, & \text{otherwise} \end{cases}$,then the probability of the waiting time being not more than $4$ minutes is = . . . . . . .

MHT CET 2016 All MHT CET

A
$0.3$
B
$0.8$
C
$0.2$
D
$0.5$

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Class 12

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Probability

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Probability distribution

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$A$ bag contains $2$ white and $1$ red balls. One ball is drawn at random and then put back in the box after noting its colour. The process is repeated again. If $X$ denotes the number of red balls recorded in the two draws,describe $X$.

Easy

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Find the mean number of heads in three tosses of a fair coin.

Medium

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Let a sample space be $S = \{\omega_{1}, \omega_{2}, \ldots, \omega_{6}\}$. Which of the following assignments of probabilities to each outcome is valid?

Outcome $\omega_1$ $\omega_2$ $\omega_3$ $\omega_4$ $\omega_5$ $\omega_6$

$(b)$ $1$ $0$ $0$ $0$ $0$ $0$

Outcome	$\omega_1$	$\omega_2$	$\omega_3$	$\omega_4$	$\omega_5$	$\omega_6$
$(b)$	$1$	$0$	$0$	$0$	$0$	$0$

Easy

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Let $X$ be the discrete random variable representing the number $(x)$ that appears on the face of a biased die when it is rolled. The probability distribution of $X$ is given by:
$X = x$ $1$ $2$ $3$ $4$ $5$ $6$
$P(X = x)$ $0.1$ $0.15$ $0.3$ $0.25$ $k$ $k$

Find the variance of $X$.

MediumTS EAMCET 2020

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If a random variable $X$ follows the Poisson distribution with variance $3$, then $P(X=r)$ is maximum, when $r=$

EasyTS EAMCET 2020

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$X = x$	$1$	$2$	$3$	$4$	$5$	$6$
$P(X = x)$	$0.1$	$0.15$	$0.3$	$0.25$	$k$	$k$

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$A$ bag contains $2$ white and $1$ red balls. One ball is drawn at random and then put back in the box after noting its colour. The process is repeated again. If $X$ denotes the number of red balls recorded in the two draws,describe $X$.

Find the mean number of heads in three tosses of a fair coin.

Let a sample space be $S = \{\omega_{1}, \omega_{2}, \ldots, \omega_{6}\}$. Which of the following assignments of probabilities to each outcome is valid? Outcome $\omega_1$ $\omega_2$ $\omega_3$ $\omega_4$ $\omega_5$ $\omega_6$ $(b)$ $1$ $0$ $0$ $0$ $0$ $0$

Let $X$ be the discrete random variable representing the number $(x)$ that appears on the face of a biased die when it is rolled. The probability distribution of $X$ is given by:$X = x$$1$$2$$3$$4$$5$$6$$P(X = x)$$0.1$$0.15$$0.3$$0.25$$k$$k$Find the variance of $X$.

If a random variable $X$ follows the Poisson distribution with variance $3$, then $P(X=r)$ is maximum, when $r=$

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Let a sample space be $S = \{\omega_{1}, \omega_{2}, \ldots, \omega_{6}\}$. Which of the following assignments of probabilities to each outcome is valid?

Outcome $\omega_1$ $\omega_2$ $\omega_3$ $\omega_4$ $\omega_5$ $\omega_6$

$(b)$ $1$ $0$ $0$ $0$ $0$ $0$

Let $X$ be the discrete random variable representing the number $(x)$ that appears on the face of a biased die when it is rolled. The probability distribution of $X$ is given by:
$X = x$ $1$ $2$ $3$ $4$ $5$ $6$
$P(X = x)$ $0.1$ $0.15$ $0.3$ $0.25$ $k$ $k$

Find the variance of $X$.