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Continuous and Standard Probability Distributions Short Notes for GATE CS

Continuous and Standard Probability Distributions short notes for GATE CS: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus solved pr

continuous and standard probability distributions short notes

Quick Recap: Exponential Distribution

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Quick Recap: Identification Checklist

Identification Checklist Form Check Is it C \cdot e^{- k (x - μ)^{2}} ? Extract μ From (x - μ)^{2} Extract σ^{2} From k = \frac{1}{2 σ ^{2}} ⟹ σ = \frac{1}{2 k} Validate C C = \frac{1}{σ 2 π} If all steps hold, X \sim N (μ, σ^{2})

Quick Recap: The 2-Step Exam Checklist

Quick Recap: The 2-Step Exam Checklist Find C Set \int f (x) d x = 1 over the valid domain. Find Probability Integrate f (x) over the target interval, clipping the bounds to the valid domain. CDF Shortcut If the cumulative function F (x) is given, just calculate F (b) - F (a) . If all steps hold, your probability model is fully normalized and ready for interval calculations.

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Question 1

Level 1: Warm-up

For a normal random variable X \sim N (0, 9), the standard deviation of X is ______.

Question 2

Level 1: Warm-up

The mean lifetime of a component is 4 hours. If the lifetime follows an exponential distribution, the variance of the lifetime (in hours^{2}) is ______.

Question 3

Level 1: Warm-up

The probability density function of a random variable X is given by f (x) = \frac{1}{σ 2 π} e^{- \frac{( x - μ ) ^{2}}{2 σ ^{2}}} . This represents which distribution?

Question 4

Level 1: Warm-up

Let X be an exponential random variable with parameter λ = 4 . The value of the probability density function f (x) at x = 0 is ______.

Question 5

Level 1: Warm-up

Given the probability density function f (x) = C \cdot exp (- \frac{( x - 5 ) ^{2}}{50}) for x \in (- \infty, \infty), the mean of the random variable X is ______.

Question 6

Level 1: Warm-up

For a normal random variable X \sim N (5, 16), the variance of X is ______.

Question 7

Level 1: Warm-up

Let X be an exponential random variable with parameter λ = 5 . The value of the probability density function f (x) at x = - 1 is ______.

Question 8

Level 1: Warm-up

A continuous random variable X has PDF f (x) = C x for 0 \leq x \leq 2 and 0 otherwise. The value of the constant C is ______.

Question 9

Level 1: Warm-up

The maximum value of the probability density function f (x) = 2 e^{- 2 x} for x \geq 0 is ______.

Question 10

Level 1: Warm-up

Let f (x) = 2 x for 0 \leq x \leq 1 and 0 otherwise. The probability P (0.5 \leq X \leq 1) is ______.

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Continuous and Standard Probability Distributions Short Notes for GATE CS

Continuous and Standard Probability Distributions short notes for GATE CS: 3 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Quick Recap: Exponential Distribution

Summary Checklist

PDF: $f (x) = λ e^{- λ x}$ for $x \geq 0$
CDF: $F (x) = 1 - e^{- λ x}$
Mean: $E [X] = \frac{1}{λ}$
Survival: $P (X > t) = e^{- λ t}$
Memoryless: Past survival doesn't affect future
Key Fact: $P (X > E [X]) = e^{- 1} \approx 0.37$

Quick Recap: Identification Checklist

Identification Checklist

Form Check

Is it $C \cdot e^{- k (x - μ)^{2}}$ ?

Extract $μ$

From $(x - μ)^{2}$

Extract $σ^{2}$

From $k = \frac{1}{2 σ ^{2}} ⟹ σ = \frac{1}{2 k}$

Validate $C$

$C = \frac{1}{σ 2 π}$

If all steps hold, $X \sim N (μ, σ^{2})$

Quick Recap: The 2-Step Exam Checklist

Find $C$

Set $\int f (x) d x = 1$ over the valid domain.

Find Probability

Integrate $f (x)$ over the target interval, clipping the bounds to the valid domain.

CDF Shortcut

If the cumulative function $F (x)$ is given, just calculate $F (b) - F (a)$ .

If all steps hold, your probability model is fully normalized and ready for interval calculations.

Continuous and Standard Probability Distributions: Solved Questions with Step-by-Step Explanations (10 Problems)

Question 1 · Engineering Mathematics MCQ

For a normal random variable $X \sim N (0, 9)$ , the standard deviation of $X$ is ______.

A.
9
B.
3
C.
81
D.
\sqrt{3}

Correct Answer:

Step-by-Step Solution

Key idea: In the notation $N (μ, σ^{2})$ , the second parameter is the variance, not the standard deviation.

Step 1: Recall the notation. $X \sim N (μ, σ^{2})$ means:

Mean = $μ$
Variance = $σ^{2}$
Standard deviation = $σ = σ^{2}$

Step 2: Given $X \sim N (0, 9)$ :

Variance = $σ^{2} = 9$
Standard deviation = $σ = 9 = 3$

Answer: 3

Question 2 · Engineering Mathematics MCQ

The mean lifetime of a component is 4 hours. If the lifetime follows an exponential distribution, the variance of the lifetime (in hours $^{2}$ ) is ______.

A.
4
B.
8
C.
16
D.
2

Correct Answer:

Step-by-Step Solution

Key idea: For an exponential distribution, the variance equals the square of the mean.

Step 1: Recall the relationship. For $X \sim Exp (λ)$ :

$E [X] = \frac{1}{λ}, Var (X) = \frac{1}{λ ^{2}} = (E [X])^{2}$

Step 2: Given $E [X] = 4$ hours, compute the variance.

$Var (X) = 4^{2} = 16 hours^{2}$

Answer: 16

Question 3 · Engineering Mathematics MCQ

The probability density function of a random variable $X$ is given by $f (x) = \frac{1}{σ 2 π} e^{- \frac{( x - μ ) ^{2}}{2 σ ^{2}}}$ . This represents which distribution?

A.
Exponential
B.
Normal
C.
Poisson
D.
Uniform

Correct Answer:

Step-by-Step Solution

Key idea: The given formula is the standard definition of the Normal (Gaussian) probability density function.

Step 1: Identify the structure. The presence of $e^{- quadratic term}$ and the specific coefficient $\frac{1}{σ 2 π}$ are the unique signatures of the Normal distribution.

Step 2: Compare with other distributions. Exponential has $e^{- λ x}$ , Poisson is discrete, Uniform is constant. None match this form.

Answer: Normal

Question 4 · Engineering Mathematics MCQ

Let $X$ be an exponential random variable with parameter $λ = 4$ . The value of the probability density function $f (x)$ at $x = 0$ is ______.

A.
4
B.
0
C.
1
D.
0.25

Correct Answer:

Step-by-Step Solution

Key idea: The exponential PDF is $f (x) = λ e^{- λ x}$ for $x \geq 0$ .

Step 1: Identify the formula. For exponential distribution, $f (x) = λ e^{- λ x}$ .

Step 2: Substitute $x = 0$ and $λ = 4$ .

$f (0) = 4 \cdot e^{- 4 \cdot 0} = 4 \cdot e^{0} = 4 \cdot 1 = 4$

Answer: 4

Question 5 · Engineering Mathematics MCQ

Given the probability density function $f (x) = C \cdot exp (- \frac{( x - 5 ) ^{2}}{50})$ for $x \in (- \infty, \infty)$ , the mean of the random variable $X$ is ______.

A.
50
B.
25
C.
5
D.
10

Correct Answer:

Step-by-Step Solution

Key idea: In the normal PDF form $f (x) = C \cdot exp (- k (x - μ)^{2})$ , the mean $μ$ is the value that makes the squared term zero.

Step 1: Compare with the standard normal PDF form:

$f (x) = \frac{1}{σ 2 π} exp (- \frac{( x - μ ) ^{2}}{2 σ ^{2}})$

Step 2: Match the exponent. Given:

$- \frac{( x - 5 ) ^{2}}{50}$

This matches $- \frac{( x - μ ) ^{2}}{2 σ ^{2}}$ with $μ = 5$ .

Step 3: The mean is $μ = 5$ .

Answer: 5

Question 6 · Engineering Mathematics MCQ

For a normal random variable $X \sim N (5, 16)$ , the variance of $X$ is ______.

A.
5
B.
4
C.
16
D.
256

Correct Answer:

Step-by-Step Solution

Key idea: In the notation $N (μ, σ^{2})$ , the second parameter represents the variance directly.

Step 1: Recall the notation convention. $X \sim N (μ, σ^{2})$ means Mean = $μ$ and Variance = $σ^{2}$ .

Step 2: Extract the values. Given $N (5, 16)$ :

$μ = 5$
$σ^{2} = 16$

Step 3: The question asks for variance, which is 16.

Answer: 16

Question 7 · Engineering Mathematics MCQ

Let $X$ be an exponential random variable with parameter $λ = 5$ . The value of the probability density function $f (x)$ at $x = - 1$ is ______.

A.
5e^5
B.
0
C.
5e^{-5}
D.
1

Correct Answer:

Step-by-Step Solution

Key idea: The exponential PDF is zero for $x < 0$ .

Step 1: Recall the piecewise definition:

$f (x) = {λ e^{- λ x} 0 x \geq 0 x < 0$

Step 2: Since $x = - 1 < 0$ , we use the second case.

$f (- 1) = 0$

Answer: 0

Question 8 · Engineering Mathematics MCQ

A continuous random variable $X$ has PDF $f (x) = C x$ for $0 \leq x \leq 2$ and $0$ otherwise. The value of the constant $C$ is ______.

A.
1/2
B.
1
C.
2
D.
1/4

Correct Answer:

Step-by-Step Solution

Key idea: Use the normalization condition $\int f (x) d x = 1$ to solve for the unknown constant.

Step 1: Set up the integral over the non-zero domain $[0, 2]$ .

$\int_{0}^{2} C x d x = 1$

Step 2: Evaluate the integral.

$C [\frac{x ^{2}}{2}]_{0}^{2} = 1$

$C (\frac{4}{2} - 0) = 1$

$2 C = 1$

Step 3: Solve for C.

$C = \frac{1}{2}$

Answer: 1/2

Question 9 · Engineering Mathematics MCQ

The maximum value of the probability density function $f (x) = 2 e^{- 2 x}$ for $x \geq 0$ is ______.

A.
2
B.
1
C.
0
D.
e^{-2}

Correct Answer:

Step-by-Step Solution

Key idea: The exponential PDF is a decreasing function, so its maximum occurs at the left boundary $x = 0$ .

Step 1: Recognize that $f (x) = 2 e^{- 2 x}$ is a decreasing exponential function for $x \geq 0$ .

Step 2: The maximum value occurs at the smallest $x$ , which is $x = 0$ .

$f (0) = 2 e^{- 2 \cdot 0} = 2 e^{0} = 2 \cdot 1 = 2$

Answer: 2

Question 10 · Engineering Mathematics MCQ

Let $f (x) = 2 x$ for $0 \leq x \leq 1$ and $0$ otherwise. The probability $P (0.5 \leq X \leq 1)$ is ______.

A.
0.25
B.
0.5
C.
0.75
D.
1

Correct Answer:

Step-by-Step Solution

Key idea: Calculate the area under the PDF curve between the specified limits.

Step 1: Set up the definite integral for the interval $[0.5, 1]$ .

$P (0.5 \leq X \leq 1) = \int_{0.5}^{1} 2 x d x$

Step 2: Evaluate the integral.

$[x^{2}]_{0.5}^{1}$

Step 3: Substitute limits.

$1^{2} - (0.5)^{2} = 1 - 0.25 = 0.75$

Answer: 0.75

continuous and standard probability distributions short notes

Quick Recap: Exponential Distribution

Summary Checklist

Quick Recap: Identification Checklist

Identification Checklist

Quick Recap: The 2-Step Exam Checklist

Quick Recap: The 2-Step Exam Checklist

Try a question

Login to view the complete short notes

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

Continuous and Standard Probability Distributions Short Notes for GATE CS

Quick Recap: Exponential Distribution

Summary Checklist

Quick Recap: Identification Checklist

Identification Checklist

Quick Recap: The 2-Step Exam Checklist

Quick Recap: The 2-Step Exam Checklist

Continuous and Standard Probability Distributions: Solved Questions with Step-by-Step Explanations (10 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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