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)

Consider two functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow(1, \infty)$. Both functions are differentiable at a point c . Which of the following functions is/are ALWAYS differentiable at c ? The symbol $\cdot$ denotes product and the symbol odenotes composition of functions.