Probability Distributions Questions with Solutions

The probability distribution function of a random variable $X$ is given by $f(x)=\frac{x}{18},;0\le x\le 6$ $=0,otherwise$ Then the value of $P(X>2)$ is

If, for a binomial distribution, the number of trials is $9$, the variance is $2$ and the probability of success is greater than that of failure, find the probability of both success.

A random variable X has the following Probability Mass Function: $X$: 0, 1, 2 $P\left[X=x\right]$: $q^2$, $2pq$, $p^2$ Then the variance of X is

In a Poisson distribution, if $P\left[X=3\right]=\frac{1}{4}P\left[X=4\right]$, then $P\left[X=5\right]=kP\left[X=7\right]$, where $k$ equals

The scores of students in a national level examination are normally distributed with a mean of 500 and a standard deviation of 100. If the value of the cumulative distribution of the standard normal random variable at 0.5 is 0.691, then the probability that a randomly selected student scored between 450 and 500 is

The number of accidents per week in a town follows a Poisson distribution with mean 3. If the probability that there are three accidents in two weeks' time is $k{e}^{-6}$, then the value of $k$ is