Continuous Probability Distributions MCQs 2026

Normal approximation to binomial applies when

1. An is very small✓
2. Bp is exactly 0✓
3. Cnp and n(1−p) are both sufficiently large (often ≥ 5 or 10)✓
4. Dnp is near 0 only✓

The standard normal distribution Z has

1. Amean 0 and variance 1✓
2. Bmean 1 and variance 0✓
3. Cmean μ and variance σ✓
4. Duniform on 0 to 1✓

If X ~ N(μ, σ²), the standardized variable Z equals

1. AX / μ✓
2. B(X − σ) / μ✓
3. Cμ / σ✓
4. D(X − μ) / σ✓

For X ~ N(100, 25), σ equals

1. A25✓
2. B5✓
3. C100✓
4. D12.5✓

For X ~ N(100, 25), P(Z=0) corresponds to X equal to

1. A100✓
2. B25✓
3. C5✓
4. D0✓

The normal curve is

1. Abell-shaped and symmetric about μ✓
2. Bskewed right always✓
3. Cuniform✓
4. Ddiscrete✓

Total area under the normal PDF equals

1. A1✓
2. B0.5✓
3. C0✓
4. Dμ✓

For standard normal Z, approximately 68% of probability lies within

1. AZ = ±2✓
2. BZ = ±3✓
3. CZ = ±0.5✓
4. DZ = ±1✓

Approximately 95% of a normal distribution lies within

1. Aμ ± 2σ✓
2. Bμ ± 1σ✓
3. Cμ ± 3σ✓
4. Dμ only✓

Approximately 99.7% lies within

1. Aμ ± 3σ✓
2. Bμ ± 2σ✓
3. Cμ ± 1σ✓
4. Dμ ± 4σ✓

P(Z ≤ 0) for standard normal equals

1. A0✓
2. B0.5✓
3. C1✓
4. D0.68✓

If X ~ N(50, 100), P(X < 60) uses Z equal to

1. A1✓
2. B0.5✓
3. C2✓
4. D10✓

If X ~ N(200, 64), P(X > 216) uses Z equal to

1. A1✓
2. B4✓
3. C2✓
4. D16✓

P(Z > 1.96) is approximately

1. A0.05✓
2. B0.5✓
3. C0.025✓
4. D0.975✓

A 95% confidence interval for μ uses critical value approximately

1. A±1.00 always✓
2. B±1.96 for large samples (known σ)✓
3. C±3.00 always✓
4. D0✓

The normal PDF f(x) for X ~ N(μ,σ²) is highest at

1. Ax = σ✓
2. Bx = 0 always✓
3. Cx = μ✓
4. Dx = μ + σ✓

For normal X, median and mean are

1. Amedian always above mean✓
2. Bequal (both μ)✓
3. Cmean always zero✓
4. Dunequal always✓

The uniform distribution on [a,b] has PDF

1. Ab−a everywhere✓
2. Ba constant > 1 always✓
3. C1/(b−a) on [a,b] and 0 elsewhere✓
4. Dzero on [a,b]✓

Uniform on [0,10]: E(X) equals

1. A10✓
2. B2.5✓
3. C0✓
4. D5✓

Uniform on [0,10]: Var(X) equals

1. A25✓
2. B100/12 ≈ 8.333✓
3. C10✓
4. D5✓

The exponential distribution is often used for

1. Awaiting time until an event in a Poisson process✓
2. Bcounting discrete successes in n trials✓
3. Csampling without replacement✓
4. Dhypergeometric counts✓

Exponential(λ) has mean

1. Aλ✓
2. Bλ²✓
3. C0✓
4. D1/λ✓

The memoryless property holds for

1. Anormal distribution✓
2. Buniform only✓
3. Cbinomial✓
4. Dexponential (and geometric discrete counterpart)✓

For X ~ Bin(100, 0.5), normal approximation uses μ and σ² equal to

1. A100 and 0.5✓
2. B50 and 50✓
3. C25 and 50✓
4. D50 and 25✓

Continuity correction when approximating P(X=k) for discrete X uses

1. AP(Y=k) exactly✓
2. BP(Y<k) only without adjustment✓
3. CP(k−0.5 < Y < k+0.5) for normal Y✓
4. Dno correction ever✓

Bin(100,0.5): approximate P(X ≤ 45) with continuity correction uses

1. AP(Y < 45.5) for Y ~ N(50,25)✓
2. BP(Y < 45)✓
3. CP(Y < 50)✓
4. DP(Y < 40.5)✓

For Bin(400, 0.25), normal approx: μ=100, σ=√75≈8.66. Z for X=110 uses

1. A(110−100)/8.66 only✓
2. B100/8.66✓
3. C(110−100)/75✓
4. D(110.5−100)/8.66 with continuity correction✓

Poisson(λ) can be approximated by N(λ, λ) when

1. Aλ = 0✓
2. Bλ is large (e.g., λ ≥ 10)✓
3. Cλ = 1 always✓
4. Dλ < 1 only✓

For Poisson(100), approximate P(X ≥ 115) uses normal with μ=100, σ=10 and

1. Ano correction✓
2. Bμ=50✓
3. Ccontinuity correction at 114.5✓
4. Dσ=100✓

The standard normal table typically gives

1. AP(Z = z)✓
2. BP(Z ≥ z) only without conversion✓
3. CP(X = μ)✓
4. DP(Z ≤ z)✓

To find P(Z > a), compute

1. AP(Z ≤ a)✓
2. B1 − P(Z ≤ a)✓
3. CP(Z ≤ −a)✓
4. DP(Z = a)✓

P(−1.5 < Z < 1.5) equals

1. AP(Z<1.5) + P(Z<−1.5)✓
2. B1 − P(Z<1.5)✓
3. CP(Z<1.5) − P(Z<−1.5)✓
4. DP(Z<3)✓

By symmetry P(Z < −a) equals

1. AP(Z > a)✓
2. BP(Z < a)✓
3. C0✓
4. D1 − P(Z < 0)✓

If X ~ N(μ, σ²), then aX + b (a>0) is distributed as

1. AN(μ, σ²)✓
2. BN(aμ + b, a²σ²)✓
3. CN(μ + b, σ²)✓
4. Duniform✓

Sum of independent normal variables is

1. Aalways standard normal✓
2. Bbinomial✓
3. CPoisson✓
4. Dnormal with mean and variance added✓

For X ~ N(10, 4), P(8 < X < 12) uses Z bounds

1. A0 and 1✓
2. B−2 and +2✓
3. C8 and 12✓
4. D−1 and +1✓

P(|Z| > 2) is approximately

1. A0.0228✓
2. B0.9544✓
3. C0.0456✓
4. D0.5✓

A test score X ~ N(70, 100). A score of 85 has Z-score

1. A15✓
2. B0.15✓
3. C1.5✓
4. D2.5✓

Heights ~ N(170, 25). Proportion with height > 180 cm uses Z equal to

1. A10✓
2. B1✓
3. C2✓
4. D0.4✓

Central Limit Theorem states that

1. Aany small sample is exactly normal✓
2. Bthe sample mean is approximately normal for large n (under conditions)✓
3. Cpopulation must be normal always for any n✓
4. Dvariance equals zero✓

For non-normal population with large n, distribution of X̄ is approximately

1. ABin(n,p)✓
2. BPoisson(λ)✓
3. Cuniform only✓
4. DN(μ, σ²/n)✓

Standard error of the mean is

1. Aσ / √n✓
2. Bσ × √n✓
3. Cσ²✓
4. Dσ / n✓

If σ=20 and n=100, SE equals

1. A20✓
2. B0.2✓
3. C200✓
4. D2✓

Normal probability plot is used to

1. Acompute binomial coefficients✓
2. Bassess whether data plausibly come from a normal distribution✓
3. Cfind Poisson λ exactly✓
4. Dprove independence✓

Using normal approx for Bin(20, 0.4), μ=8, σ≈2.19. P(X≥10) with correction approximates P(Y>9.5)

1. Ausing Z=(9.5−8)/2.19✓
2. BZ=(10−8)/2.19 without correction✓
3. CZ=8/2.19✓
4. DZ=0✓

Skewed data with small n should

1. Anot rely blindly on normal approximation✓
2. Balways use Z=1.96✓
3. Cignore continuity correction✓
4. Dassume σ=0✓

The PDF of N(μ,σ²) has inflection points at

1. Aμ only✓
2. Bμ ± σ✓
3. Cμ ± 2σ✓
4. D0 only✓

For continuous uniform on [a,b], P(X < c) for a < c < b equals

1. Ac/b✓
2. B(c − a)/(b − a)✓
3. C1/(b−a)✓
4. D0.5 always✓

The 90th percentile of Z is approximately

1. A1.96✓
2. B0.90✓
3. C1.28✓
4. D2.58✓

For manufacturing: bolt diameter ~ N(10, 0.01). σ=0.1 mm. P(diameter < 9.8) uses Z=

1. A−0.2✓
2. B−2✓
3. C2✓
4. D0✓