Let X be a continuous random variable whose cumulative distribution function (CDF) $F_X(x)$, for some $t$, is given as follows:

$$ F_X(x)=\left\{\begin{array}{cc} 0 & x \leq t \\ \frac{x-t}{4-t} & t \leq x \leq 4 \\ 1 & x \geq 4 \end{array}\right. $$

If the median of X is 3 , then what is the value of $t$ ?

Let $X=a Z+b$, where Z is a standard normal random variable, and $a, b$ are two unknown constants. It is given that

$$ \begin{aligned} E[X] & =1, E[(X-E[X]) Z] \\ & =-2, E\left[(X-E[X])^2\right]=4 \end{aligned} $$

Where $E[X]$ denotes the expectation of random variable X . The values of $a, b$ are:

It is given that $P(X \geq 2)=0.25$ for an exponentially distributed random variable $X$ with $E[X]=\frac{1}{\lambda}$, where $E[X]$ denotes the expectation of $X$. What is the value of $\lambda$ ? (ln denotes natural logarithm)

There are three boxes containing white balls and black balls.

Box-1 contains 2 black and 1 white balls.

Box-2 contains 1 black and 2 white balls.

Box-3 contains 3 black and 3 white balls.

In a random experiment, one of these boxes is selected, where the probability of choosing Box-1 is $\frac{1}{2}$, Box-2 is $\frac{1}{6}$, and Box-3 is $\frac{1}{3}$. A ball is drawn at random from the selected box. Given that the ball drawn is white, the probability that it is drawn from Box-2 is ____________. (Round off to two decimal places)