(B) The sum of all probabilities in a probability distribution must be equal to $1$.$\sum P(X=x) = 1$$P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) = 1$$

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(B) The sum of all probabilities in a probability distribution must be equal to $1$.$\sum P(X=x) = 1$$P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) = 1$$0.2 + k(1) + k(2) + k(6-3) + k(6-4) = 1$$0.2 + k + 2k + 3k + 2k = 1$$0.2 + 8k = 1$$8k = 0.8$$k = 0.1$We need to find the probability that the student studies for at most two hours,which is $P(X \le 2) = P(X=0) + P(X=1) + P(X=2)$.$P(X \le 2) = 0.2 + k(1) + k(2) = 0.2 + 3k$Substituting $k = 0.1$:$P(X \le 2) = 0.2 + 3(0.1) = 0.2 + 0.3 = 0.5$.

Class 12 Mathematics Probability Probability distribution

Medium

$A$ student studies for $X$ number of hours during a randomly selected school day. The probability distribution of $X$ is given by the following form,where $k$ is a constant:
$P(X=x) = \begin{cases} 0.2, & \text{if } x=0 \\ kx, & \text{if } x=1 \text{ or } 2 \\ k(6-x), & \text{if } x=3 \text{ or } 4 \\ 0, & \text{otherwise} \end{cases}$
The probability that the student studies for at most two hours is:

MHT CET 2025 All MHT CET

A
$0.1$
B
$0.5$
C
$0.3$
D
$0.7$

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If a variable takes values $0, 1, 2, ..., n$ with frequencies proportional to the binomial coefficients $^nC_0, ^nC_1, ..., ^nC_n$,find the mean of the distribution.

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$A$ random variate $X$ takes the values $0, 1, 2, 3$ and its mean is $1.3$. If $P(X=3) = 2 P(X=1)$ and $P(X=2) = 0.3$,then $P(X=0)$ is equal to:

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In a Poisson distribution, if $\frac{P(X=5)}{P(X=2)}=\frac{1}{7500}$ and $\frac{P(X=5)}{P(X=3)}=\frac{1}{500}$, then the mean of the distribution is

MediumTS EAMCET 2025

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If the probability function of a discrete random variable $X$ is $P(X=r) = r/k$ for $r = 1, 2, 3, 4, 5$, then $P(X=2 \text{ or } X=k/3)$ is equal to:

MediumTS EAMCET 2020

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The probability distribution of a discrete random variable $X$ is given by the following table:
$X = x$ $0$ $1$ $2$ $3$ $4$
$P(X = x)$ $k$ $2k$ $4k$ $2k$ $k$

Then the value of $P(X \leq 2)$ is:

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If a variable takes values $0, 1, 2, ..., n$ with frequencies proportional to the binomial coefficients $^nC_0, ^nC_1, ..., ^nC_n$,find the mean of the distribution.

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The probability distribution of a discrete random variable $X$ is given by the following table:
$X = x$ $0$ $1$ $2$ $3$ $4$
$P(X = x)$ $k$ $2k$ $4k$ $2k$ $k$

Then the value of $P(X \leq 2)$ is: