# Probability and Statistics

 Question 1
The expected number of trials for first occurrence of a "head" in a biased coin is known to be 4. The probability of first occurrence of a "head" in the second trial is ___ (Round off to 3 decimal places).
 A 0.125 B 0.188 C 0.254 D 0.564
GATE EE 2023   Engineering Mathematics
Question 1 Explanation:
Let probability of head $=\mathrm{P}$
Let probability of Tail $=q=P-1$
$\begin{array}{|c|c|c|c|c|} \hline Trial & 1 & 2 & 3 & ... \\ \hline Probability & [\mathrm{P} & \mathrm{qP} & [latex]\mathrm{q}^{2} \mathrm{P} & .... \\ \hline \end{array}$
$\therefore$ Expected No. of trail
\begin{aligned} & =\sum_{n=1}^{\infty} p+2 P q+3 q^{2} P+\ldots \\ & =P\left(1+2 q+3 q^{2}+\ldots\right) \\ & =P(1-q)^{-2} \\ & =\frac{P}{P^{2}}=\frac{1}{P} \end{aligned}

Given : Trial $=4$
$\frac{1}{P}=4 \Rightarrow P=\frac{1}{4}$
$\therefore \quad \mathrm{q}=\frac{3}{4}$

Now, probability of head for second trail. \begin{aligned} & =\mathrm{qP} \\ & =\frac{3}{4} \times \frac{1}{4}=\frac{3}{16}=0.1875 \end{aligned}
 Question 2
Let the probability density function of a random variable $x$ be given as
$f(x)=ae^{-2|x|}$
The value of $'a'$ is _________
 A 0.5 B 1 C 1.5 D 2
GATE EE 2022   Engineering Mathematics
Question 2 Explanation:
$f(x)=\left\{\begin{matrix} ae^{2x} &x \lt 0 \\ ae^{-2x} &x \gt 0 \end{matrix}\right.$
Therefore,
\begin{aligned} \int_{-\infty }^{\infty }f(x)dx&=1\\ \int_{-\infty }^{0}ae^{2x}dx+\int_{0 }^{\infty }ae^{-2x}dx&=1\\ a\left [ \left [ \frac{e^{2x}}{2} \right ]_{-\infty }^0 +[ \left [ \frac{e^{-2x}}{-2} \right ]^{\infty }_0\right ]&=1\\ a\left [ \frac{1}{2}+\frac{1}{2} \right ]&=1\\ a&=1 \end{aligned}

 Question 3
Suppose the probability that a coin toss shows "head" is p, where $0\lt p\lt 1$. The coin is tossed repeatedly until the first "head" appears. The expected number of tosses required is
 A $p/\left ( 1-p \right )$ B $\left ( 1-p \right )/p$ C $1/p$ D $1/p^{2}$
GATE EE 2021   Engineering Mathematics
Question 3 Explanation:
$F(H) = p$, let x = Number of tosses
\begin{aligned} &\begin{array}{|c|c|c|c|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 \ldots \\ \hline P(x) & p & (1-p) p & (1-p)^{2} p & & \\ \hline \end{array}\\ &\begin{aligned} E(x) &=\sum x_{i} P(x)=p+2(1-p) p+3(p)(1-p)^{2}+\ldots \\ &=p\left[1+2(1-p)+3(1-p)^{2}+\ldots\right] \\ &=p[1-(1-p)]^{-2}=\frac{1}{p} \end{aligned} \end{aligned}
 Question 4
The probability of a resistor being defective is 0.02. There are 50 such resistors in a circuit. The probability of two or more defective resistors in the circuit (round off to two decimal places)is ________
 A 0.1 B 0.26 C 0.65 D 0.85
GATE EE 2019   Engineering Mathematics
Question 4 Explanation:
\begin{aligned} p&=0.02\\ n&=50\\ \lambda &=np=50(0.02)=1\\ p(x\geq 2)&=1-p(x \lt 2)\\ &=1-(p(x=0)+p(x=1))\\ &=1-\left ( \frac{e^{-\lambda }\lambda ^0}{0!}+\frac{e^{-\lambda }\lambda ^1}{1!} \right )\\ &=1-e^{-\lambda }(1+\lambda )\\ &=1-e^{-1}(1+1)=0.26 \end{aligned}
 Question 5
The mean-square of a zero-mean random process is $\frac{kT}{C}$, where k is Boltzmann's constant, T is the absolute temperature, and C is a capacitance. The standard deviation of the random process is
 A $\frac{kT}{C}$ B $\sqrt{\frac{kT}{C}}$ C $\frac{C}{kT}$ D $\frac{\sqrt{kT}}{C}$
GATE EE 2019   Engineering Mathematics
Question 5 Explanation:
Given that,
\begin{aligned} E(x^2)&=\frac{kT}{C}\\ E(x)&=0\\ &=E(x^2)-(E(x))^2\\ Var(x)&=\frac{kT}{C}-0=\frac{kT}{C} \end{aligned}
Standard deviation $=\sqrt{\frac{kT}{C}}$

There are 5 questions to complete.